3.5: Triple Integrals
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Learning Objectives
- Recognize when a function of three variables is integrable over a rectangular box.
- Evaluate a triple integral by expressing it as an iterated integral.
- Recognize when a function of three variables is integrable over a closed and bounded region.
- Simplify a calculation by changing the order of integration of a triple integral.
- Calculate the average value of a function of three variables.
Previously, we discussed the double integral of a function \(f(x,y)\) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function \(f(x,y,z)\) of three variables over a rectangular solid box in space, \(\mathbb{R}^3\). Later in this section we extend the definition to more general regions in \(\mathbb{R}^3\).
Integrable Functions of Three Variables
We can define a rectangular box \(B\) in \(\mathbb{R}^3\) as
\[B = \big\{(x,y,z)\,|\,a \leq x \leq b, \, c \leq y \leq d, \, e \leq z \leq f \big\}. \nonumber \]
We follow a similar procedure to what we did in previously. We divide the interval \([a,b]\) into \(l\) subintervals \([x_{i-1},x_i]\) of equal length \(\Delta x\) with
\[\Delta x = \dfrac{x_i - x_{i-1}}{l}, \nonumber \]
divide the interval \([c,d]\) into \(m\) subintervals \([y_{i-1}, y_i]\) of equal length \(\Delta y\) with
\[\Delta y = \dfrac{y_j - y_{j-1}}{m}, \nonumber \]
and divide the interval \([e,f]\) into \(n\) subintervals \([z_{i-1},z_i]\) of equal length \(\Delta z\) with
\[\Delta z = \dfrac{z_k - z_{k-1}}{n} \nonumber \]
Then the rectangular box \(B\) is subdivided into \(lmn\) subboxes:
\[B_{ijk} = [x_{i-1}, x_i] \times [y_{i-1}, y_i] \times [z_{i-1},z_i], \nonumber \]
as shown in Figure \(\PageIndex{1}\).
For each \(i, \, j,\) and \(k\), consider a sample point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) in each sub-box \(B_{ijk}\). We see that its volume is \(\Delta V = \Delta x \Delta y \Delta z\). Form the triple Riemann sum
\[\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f ( x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\,\Delta x \Delta y \Delta z. \nonumber \]
We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.
Definition: The triple integral
The triple integral of a function \(f(x,y,z)\) over a rectangular box \(B\) is defined as
\[\lim_{l,m,n\rightarrow\infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f ( x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\,\Delta x \Delta y \Delta z = \iiint_B f(x,y,z) \,dV \nonumber \] if this limit exists.
When the triple integral exists on \(B\) the function \(f(x,y,z)\) is said to be integrable on \(B\). Also, the triple integral exists if \(f(x,y,z)\) is continuous on \(B\). Therefore, we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, \(f\) is bounded on \(B\) and continuous except possibly on the boundary of \(B\). The sample point \((x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)\) can be any point in the rectangular sub-box \(B_{ijk}\) and all the properties of a double integral apply to a triple integral. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections.
Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s theorem for triple integrals exists.
Fubini’s Theorem for Triple Integrals
If \(f(x,y,z)\) is continuous on a rectangular box \(B = [a,b] \times [c,d] \times [e,f]\), then
\[\iint_B f(x,y,z) \,dV = \int_e^f \int_c^d \int_a^b f(x,y,z) \,dx \, dy \, dz. \nonumber \]
This integral is also equal to any of the other five possible orderings for the iterated triple integral.
For \(a, b, c, d, e\) and \(f\) real numbers, the iterated triple integral can be expressed in six different orderings:
\[\begin{align} \int_e^f \int_c^d \int_a^b f(x,y,z)\, dx \, dy \, dz = \int_e^f \left( \int_c^d \left( \int_a^b f(x,y,z) \,dx \right) dy \right) dz \\ = \int_c^d \left( \int_e^f \left( \int_a^b f(x,y,z) \,dx \right)dz \right) dy \\ = \int_a^b \left( \int_e^f \left( \int_c^d f(x,y,z) \,dy \right)dz \right) dx \\ = \int_e^f \left( \int_a^b \left( \int_c^d f(x,y,z) \,dy \right) dx \right) dz \\ = \int_c^d \left( \int_a^b \left( \int_c^d f(x,y,z) \,dz\right)dx \right) dy \\ = \int_a^b \left( \int_c^d \left( \int_e^f f(x,y,z) \,dz \right) dy \right) dx \end{align} \nonumber \]
For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini’s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).
Example \(\PageIndex{1}\): Evaluating a Triple Integral
Evaluate the triple integral \[\int_{z=0}^{z=1} \int_{y=2}^{y=4} \int_{x=-1}^{x=5} (x + yz^2)\, dx \, dy \, dz. \nonumber \]
Solution
The order of integration is specified in the problem, so integrate with respect to \(x\) first, then y , and then \(z\).
\[\begin{align*}&\int_{z=0}^{z=1} \int_{y=2}^{y=4} \int_{x=-1}^{x=5} (x + yz^2) \,dx \,dy \,dz \\ &= \int_{z=0}^{z=1} \int_{y=2}^{y=4} \left. \left[ \dfrac{x^2}{2} + xyz^2\right|_{x=-1}^{x=5}\right]\,dy \,dz &&\text{Integrate with respect to $x$.}\\ &= \int_{z=0}^{z=1} \int_{y=2}^{y=4} \left[12+6yz^2\right] \,dy \,dz &&\text{Evaluate.}\\ &= \int_{z=0}^{z=1} \left[ \left.12y+6\dfrac{y^2}{2}z^2 \right|_{y=2}^{y=4} \right] dz &&\text{Integrate with respect to $y$.} \\ &= \int_{z=0}^{z=1} [24+36z^2] \, dz &&\text{Evaluate.} \\ &= \left[ 24z+36\dfrac{z^3}{3} \right]_{z=0}^{z=1} &&\text{Integrate with respect to $z$.}\\ &=36. &&\text{Evaluate.}\end{align*}\]
Example \(\PageIndex{2}\): Evaluating a Triple Integral
Evaluate the triple integral
\[\iiint_B x^2 yz \,dV \nonumber \]
where \(B = \big\{(x,y,z)\,|\, - 2 \leq x \leq 1, \, 0 \leq y \leq 3, \, 1 \leq z \leq 5 \big\} \) as shown in Figure \(\PageIndex{2}\).
Solution
The order is not specified, but we can use the iterated integral in any order without changing the level of difficulty. Choose, say, to integrate \(y\) first, then \(x\), and then \(z\).
\[\begin{align*}\iiint\limits_{B} x^2 yz \,dV &= \int_1^5 \int_{-2}^1 \int_0^3 [x^2 yz] \,dy \, dx \, dz \\&= \int_1^5 \int_{-2}^1 \left[ \left. x^2 \dfrac{y^3}{3} z\right|_0^3 \right] dx \, dz \\&= \int_1^5 \int_{-2}^1 \dfrac{y}{2} x^2 z \,dx \, dz \\&= \int_1^5 \left[ \left. \dfrac{9}{2} \dfrac{x^3}{3} z \right|_{-2}^1 \right] dz \\&= \int_1^5 \dfrac{27}{2} z \, dz \\&= \left. \dfrac{27}{2} \dfrac{z^2}{2} \right|_1^5 = 162.\end{align*}\]
Now try to integrate in a different order just to see that we get the same answer. Choose to integrate with respect to \(x\) first, then \(z\), then \(y\)
\[\begin{align*}\iiint\limits_{B} x^2yz \,dV &= \int_0^3 \int_1^5 \int_{-2}^1 [x^2yz] \,dx\, dz\, dy \\&= \int_0^3 \int_1^5 \left[ \left. \dfrac{x^3}{3} yz \right|_{-2}^1 \right] dz \,dy \\&= \int_0^3 \int_1^5 3yz \; dz \,dy \\&= \int_0^3 \left.\left[ 3y\dfrac{z^2}{2} \right|_1^5 \right] \,dy \\&= \int_0^3 36y \; dy \\&= \left. 36\dfrac{y^2}{2} \right|_0^3 =18(9-0) =162.\end{align*}\]
Exercise \(\PageIndex{1}\)
Evaluate the triple integral
\[\iiint_B z \, \sin \, x \, \cos \, y \, dV\nonumber \]
where \(B = \big\{(x,y,z)\,|\,0 \leq x \leq \pi, \, \dfrac{3\pi}{2} \leq y \leq 2\pi, \, 1 \leq z \leq 3 \big\}\).
- Hint
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Follow the steps in the previous example.
- Answer
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\[\iiint_B z \, \sin \, x \, \cos \, y \, dV = 8 \nonumber \]
Triple Integral over a General Region
The triple integral of a continuous function \(f(x,y,z)\) over a general three-dimensional region
\[E = \big\{(x,y,z)\,|\,(x,y) \in D, \, u_1(x,y) \leq z \leq u_2(x,y) \big\} \nonumber \]
in \(\mathbb{R}^3\), where \(D\) is the projection of \(E\) onto the \(xy\)-plane, is
\[\iiint_E f(x,y,z) \,dV = \iint_D \left[\int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) \,dz \right] \, dA. \nonumber \]
Similarly, we can consider a general bounded region \(D\) in the \(xy\)-plane and two functions \(y = u_1(x,z)\) and \(y = u_2(x,z)\) such that \(u_1(x,z) \leq u_2(x,z)\) for all \((x,z)\) in \(D\). Then we can describe the solid region \(E\) in \(\mathbb{R}^3\) as
\[E = \big\{(x,y,z)\,|\,(x,z) \in D, \, u_1(x,z) \leq z \leq u_2(x,z) \big\} \nonumber \] where \(D\) is the projection of \(E\) onto the \(xy\)-plane and the triple integral is
\[\iiint_E f(x,y,z)\,dV = \iint_D \left[\int_{u_1(x,z)}^{u_2(x,z)} f(x,y,z) \,dy \right] \, dA. \nonumber \]
Finally, if \(D\) is a general bounded region in the \(xy\)-plane and we have two functions \(x = u_1(y,z)\) and \(x = u_2(y,z)\) such that \(u_1(y,z) \leq u_2(y,z)\) for all \((y,z)\) in \(D\), then the solid region \(E\) in \(\mathbb{R}^3\) can be described as
\[E = \big\{(x,y,z)\,|\,(y,z) \in D, \, u_1(y,z) \leq z \leq u_2(y,z) \big\} \nonumber \] where \(D\) is the projection of \(E\) onto the \(xy\)-plane and the triple integral is
\[\iiint_E f(x,y,z)\,dV = \iint_D \left[\int_{u_1(y,z)}^{u_2(y,z)} f(x,y,z) \, dx \right] \, dA. \nonumber \]
Note that the region \(D\) in any of the planes may be of Type I or Type II as described in previously. If \(D\) in the \(xy\)-plane is of Type I (Figure \(\PageIndex{4}\)), then
\[E = \big\{(x,y,z)\,|\,a \leq x \leq b, \, g_1(x) \leq y \leq g_2(x), \, u_1(x,y) \leq z \leq u_2(x,y) \big\}. \nonumber \]
Then the triple integral becomes
\[\iiint_E f(x,y,z) \,dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) \,dz \, dy \, dx. \nonumber \]
If \(D\) in the \(xy\)-plane is of Type II (Figure \(\PageIndex{5}\)), then
\[E = \big\{(x,y,z)\,|\,c \leq x \leq d, h_1(x) \leq y \leq h_2(x), \, u_1(x,y) \leq z \leq u_2(x,y) \big\}. \nonumber \]
Then the triple integral becomes
\[\iiint_E f(x,y,z) \,dV = \int_{y=c}^{y=d} \int_{x=h_1(y)}^{x=h_2(y)} \int_{z=u_1(x,y)}^{z=u_2(x,y)} f(x,y,z)\,dz \, dx \, dy. \nonumber \]
Example \(\PageIndex{3A}\): Evaluating a Triple Integral over a General Bounded Region
Evaluate the triple integral of the function \(f(x,y,z) = 5x - 3y\) over the solid tetrahedron bounded by the planes \(x = 0, \, y = 0, \, z = 0\), and \(x + y + z = 1\).
Solution
Figure \(\PageIndex{6}\) shows the solid tetrahedron \(E\) and its projection \(D\) on the \(xy\)-plane.
We can describe the solid region tetrahedron as
\[E = \big\{(x,y,z)\,|\,0 \leq x \leq 1, \, 0 \leq y \leq 1 - x, \, 0 \leq z \leq 1 - x - y \big\}. \nonumber \]
Hence, the triple integral is
\[\iiint_E f(x,y,z) \,dV = \int_{x=0}^{x=1} \int_{y=0}^{y=1-x} \int_{z=0}^{z=1-x-y} (5x - 3y) \,dz \, dy \, dx. \nonumber \]
To simplify the calculation, first evaluate the integral \(\displaystyle \int_{z=0}^{z=1-x-y} (5x - 3y) \,dz\). We have
\[\int_{z=0}^{z=1-x-y} (5x - 3y) \,dz = (5x - 3y)z \bigg|_{z=0}^{z=1-x-y} = (5x - 3y)(1 - x - y).\nonumber \]
Now evaluate the integral
\[\int_{y=0}^{y=1-x} (5x - 3y)(1 - x - y) \,dy, \nonumber \]
obtaining
\[\int_{y=0}^{y=1-x} (5x - 3y)(1 - x - y)\,dy = \dfrac{1}{2}(x - 1)^2 (6x - 1).\nonumber \]
Finally evaluate
\[\int_{x=0}^{x=1} \dfrac{1}{2}(x - 1)^2 (6x - 1)\,dx = \dfrac{1}{12}.\nonumber \]
Putting it all together, we have
\[\iiint_E f(x,y,z)\,dV = \int_{x=0}^{x=1} \int_{y=0}^{y=1-x} \int_{z=0}^{z=1-x-y}(5x - 3y)\,dz \, dy \, dx = \dfrac{1}{12}.\nonumber \]
Just as we used the double integral \[\iint_D 1 \,dA \nonumber \] to find the area of a general bounded region \(D\) we can use \[\iiint_E 1\,dV \nonumber \] to find the volume of a general solid bounded region \(E\). The next example illustrates the method.
Example \(\PageIndex{3B}\): Finding a Volume by Evaluating a Triple Integral
Find the volume of a right pyramid that has the square base in the \(xy\)-plane \([-1,1] \times [-1,1]\) and vertex at the point \((0, 0, 1)\) as shown in the following figure.
Solution
In this pyramid the value of \(z\) changes from 0 to 1 and at each height \(z\) the cross section of the pyramid for any value of \(z\) is the square
\[[-1 + z, \, 1 - z] \times [-1 + z, \, 1 - z].\nonumber \]
Hence, the volume of the pyramid is \[\iiint_E 1\,dV\nonumber \] where
\[E = \big\{(x,y,z)\,|\,0 \leq z \leq 1, \, -1 + z \leq y \leq 1 - z, \, -1 + z \leq x \leq 1 - z \big\}.\nonumber \]
Thus, we have
\[\begin{align*} \iiint_E 1\,dV &= \int_{z=0}^{z=1} \int_{y=-1+z}^{y=1-z} \int_{x=-1+z}^{x=1-z} 1\,dx \, dy \, dz \\[5pt]
&= \int_{z=0}^{z=1} \int_{y=-1+z}^{y=1-z} (2 - 2z)\, dy \, dz \\[5pt]
&= \int_{z=0}^{z=1}(2 - 2z)^2 \,dz = \dfrac{4}{3}. \end{align*}\]
Hence, the volume of the pyramid is \(\dfrac{4}{3}\) cubic units.
Exercise \(\PageIndex{3}\)
Consider the solid sphere \(E = \big\{(x,y,z)\,|\,x^2 + y^2 + z^2 = 9 \big\}\). Write the triple integral \[\iiint_E f(x,y,z) \,dV\nonumber \] for an arbitrary function \(f\) as an iterated integral. Then evaluate this triple integral with \(f(x,y,z) = 1\). Notice that this gives the volume of a sphere using a triple integral.
- Hint
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Follow the steps in the previous example. Use symmetry.
- Answer
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\[\begin{align*} \iiint_E 1\,dV = 8 \int_{x=-3}^{x=3} \int_{y=-\sqrt{9-z^2}}^{y=\sqrt{9-z^2}}\int_{z=-\sqrt{9-x^2-y^2}}^{z=\sqrt{9-x^2-y^2}} 1\,dz \, dy \, dx \\ = 36 \pi \,\text{cubic units}. \end{align*}\]
Changing the Order of Integration
As we have already seen in double integrals over general bounded regions, changing the order of the integration is done quite often to simplify the computation. With a triple integral over a rectangular box, the order of integration does not change the level of difficulty of the calculation. However, with a triple integral over a general bounded region, choosing an appropriate order of integration can simplify the computation quite a bit. Sometimes making the change to polar coordinates can also be very helpful. We demonstrate two examples here.
Example \(\PageIndex{4}\): Changing the Order of Integration
Consider the iterated integral
\[\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y} f(x,y,z)\,dz \, dy \, dx. \nonumber \]
The order of integration here is first with respect to z , then y , and then x . Express this integral by changing the order of integration to be first with respect to \(x\), then \(z\), and then \(y\). Verify that the value of the integral is the same if we let \(f (x,y,z) =xyz\).
Solution
The best way to do this is to sketch the region \(E\) and its projections onto each of the three coordinate planes. Thus, let
\[E = \big\{(x,y,z)\,|\,0 \leq x \leq 1, \, 0 \leq y \leq x^2, \, 0 \leq z \leq y \big\}.\nonumber \]
and
\[\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=x^2} f(x,y,z) \,dz \, dy \, dx = \iiint_E f(x,y,z)\,dV.\nonumber \]
We need to express this triple integral as
\[\int_{y=c}^{y=d} \int_{z=v_1(y)}^{z=v_2(y)} \int_{x=u_1(y,z)}^{x=u_2(y,z)} f(x,y,z)\,dx \, dz \, dy.\nonumber \]
Knowing the region \(E\) we can draw the following projections (Figure \(\PageIndex{8}\)):
on the \(xy\)-plane is \(D_1 = \big\{(x,y)\,|\, 0 \leq x \leq 1, \, 0 \leq y \leq x^2 \big\} = \{ (x,y) \,|\, 0 \leq y \leq 1, \, \sqrt{y} \leq x \leq 1 \big\},\)
on the \(yz\)-plane is \(D_2 = \big\{(y,z) \,|\, 0 \leq y \leq 1, \, 0 \leq z \leq y^2 \big\}\), and
on the \(xz\)-plane is \(D_3 = \big\{(x,z) \,|\, 0 \leq x \leq 1, \, 0 \leq z \leq x^2 \big\}\).
Now we can describe the same region \(E\) as \(\big\{(x,y,z) \,|\, 0 \leq y \leq 1, \, 0 \leq z \leq y^2, \, \sqrt{y} \leq x \leq 1 \big\}\), and consequently, the triple integral becomes
\[\int_{y=c}^{y=d} \int_{z=v_1(y)}^{z=v_2(y)} \int_{x=u_1(y,z)}^{x=u_2(y,z)} f(x,y,z)\,dx \, dz \, dy = \int_{y=0}^{y=1} \int_{z=0}^{z=x^2} \int_{x=\sqrt{y}}^{x=1} f(x,y,z)\,dx \, dz \, dy \nonumber \]
Now assume that \(f (x,y,z) = xyz\) in each of the integrals.
Exercise \(\PageIndex{4}\)
Write five different iterated integrals equal to the given integral
\[\int_{z=0}^{z=4} \int_{y=0}^{y=4-z} \int_{x=0}^{x=\sqrt{y}} f(x,y,z) \, dx \, dy \, dz.\nonumber \]
- Hint
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Follow the steps in the previous example, using the region \(E\) as \( \big\{(x,y,z) \,|\, 0 \leq z \leq 4, \, 0 \leq y \leq 4 - z, \, 0 \leq x \leq \sqrt{y} \big\}\), and describe and sketch the projections onto each of the three planes, five different times.
- Answer
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\[(i) \, \int_{z=0}^{z=4} \int_{x=0}^{x=\sqrt{4-z}} \int_{y=x^2}^{y=4-z} f(x,y,z) \, dy \, dx \, dz, \, (ii) \, \int_{y=0}^{y=4} \int_{z=0}^{z=4-y} \int_{x=0}^{x=\sqrt{y}} f(x,y,z) \,dx \, dz \, dy, \,(iii) \, \int_{y=0}^{y=4} \int_{x=0}^{x=\sqrt{y}} \int_{z=0}^{Z=4-y} f(x,y,z) \,dz \, dx \, dy, \, \nonumber \]
\[ (iv) \, \int_{x=0}^{x=2} \int_{y=x^2}^{y=4} \int_{z=0}^{z=4-y} f(x,y,z) \,dz \, dy \, dx, \, (v) \int_{x=0}^{x=2} \int_{z=0}^{z=4-x^2} \int_{y=x^2}^{y=4-z} f(x,y,z) \,dy \, dz \, dx \nonumber \]
Example \(\PageIndex{5}\): Changing Integration Order and Coordinate Systems
Evaluate the triple integral
\[\iiint_{E} \sqrt{x^2 + z^2} \,dV, \nonumber \]
where \(E\) is the region bounded by the paraboloid \(y = x^2 + z^2\) (Figure \(\PageIndex{9}\)) and the plane \(y = 4\).
Solution
The projection of the solid region \(E\) onto the \(xy\)-plane is the region bounded above by \(y = 4\) and below by the parabola \(y = x^2\) as shown.
Thus, we have
\[E = \big\{(x,y,z) \,|\, -2 \leq x \leq 2, \, x^2 \leq y \leq 4, \, -\sqrt{y - x^2} \leq z \leq \sqrt{y - x^2} \big\}.\nonumber \]
The triple integral becomes
\[\iiint_E \sqrt{x^2 + z^2} \,dV = \int_{x=-2}^{x=2} \int_{y=x^2}^{y=4} \int_{z=-\sqrt{y-x^2}}^{z=\sqrt{y-x^2}} \sqrt{x^2 + z^2} \,dz \, dy \, dx.\nonumber \]
This expression is difficult to compute, so consider the projection of \(E\) onto the \(xz\)-plane. This is a circular disc \(x^2 + z^2 \leq 4\). So we obtain
\[\iiint_E \sqrt{x^2 + z^2} \,dV = \int_{x=-2}^{x=2} \int_{y=x^2}^{y=4} \int_{z=-\sqrt{y-x^2}}^{z=\sqrt{y-x^2}} \sqrt{x^2 + z^2} \,dz \, dy \, dx = \int_{x=-2}^{x=2} \int_{z=-\sqrt{4-x^2}}^{z=\sqrt{4-x^2}} \int_{y=x^2+z^2}^{y=4} \sqrt{x^2 + z^2} \,dy \, dz \, dx.\nonumber \]
Here the order of integration changes from being first with respect to \(z\) then \(y\) and then \(x\) to being first with respect to \(y\) then to \(z\) and then to \(x\). It will soon be clear how this change can be beneficial for computation. We have
\[\int_{x=-2}^{x=2} \int_{z=\sqrt{4-x^2}}^{z=\sqrt{4-x^2}} \int_{y=x^2+z^2}^{y=4} \sqrt{x^2 + z^2} \,dy \, dz \, dx = \int_{x=-2}^{x=2} \int_{z=-\sqrt{4-x^2}}^{z=\sqrt{4-x^2}} (4 - x^2 - z^2) \sqrt{x^2 + z^2} \,dz \, dx.\nonumber \]
Now use the polar substitution \(x = r \, \cos \, \theta, \, z = r \, \sin \, \theta\), and \(dz \, dx = r \, dr \, d\theta\) in the \(xz\)-plane. This is essentially the same thing as when we used polar coordinates in the \(xy\)-plane, except we are replacing \(y\) by \(z\). Consequently the limits of integration change and we have, by using \(r^2 = x^2 + z^2\),
\[\int_{x=-2}^{x=2} \int_{z=-\sqrt{4-x^2}}^{z=\sqrt{4-x^2}} (4 - x^2 - z^2) \sqrt{x^2 + z^2}\,dz \, dx = \int_{\theta=0}^{\theta=2\pi} \int_{r=0}^{r=2} (4 - r^2) rr \, dr \, d\theta = \int_0^{2\pi} \left. \left[ \dfrac{4r^3}{3} - \dfrac{r^5}{5} \right|_0^2 \right] \, d\theta = \int_0^{2\pi} \dfrac{64}{15} \,d\theta = \dfrac{128\pi}{15}\nonumber \]
Average Value of a Function of Three Variables
Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.
Average Value of a Function of Three Variables
If \(f(x,y,z)\) is integrable over a solid bounded region \(E\) with positive volume \(V \, (E),\) then the average value of the function is
\[f_{ave} = \dfrac{1}{V \, (E)} \iiint_E f(x,y,z) \, dV. \nonumber \]
Note that the volume is
\[V \, (E) = \iiint_E 1 \,dV. \nonumber \]
Example \(\PageIndex{6}\): Finding an Average Temperature
The temperature at a point \((x,y,z)\) of a solid \(E\) bounded by the coordinate planes and the plane \(x + y + z = 1\) is \(T(x,y,z) = (xy + 8z + 20) \, \text{°}\text{C} \). Find the average temperature over the solid.
Solution
Use the theorem given above and the triple integral to find the numerator and the denominator. Then do the division. Notice that the plane \(x + y + z = 1\) has intercepts \((1,0,0), \, (0,1,0),\) and \((0,0,1)\). The region \(E\) looks like
\[E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 1 - x, \, 0 \leq z \leq 1 - x - y \big\}.\nonumber \]
Hence the triple integral of the temperature is
\[\iiint_E f(x,y,z) \,dV = \int_{x=0}^{x=1} \int_{y=0}^{y=1-x} \int_{z=0}^{z=1-x-y} (xy + 8z + 20) \, dz \, dy \, dx = \dfrac{147}{40}. \nonumber \]
The volume evaluation is
\[V \, (E) = \iiint_E 1\,dV = \int_{x=0}^{x=1} \int_{y=0}^{y=1-x} \int_{z=0}^{z=1-x-y} 1 \,dz \, dy \, dx = \dfrac{1}{6}. \nonumber \]
Hence the average value is
\[ T_{ave} = \dfrac{147/40}{1/6} = \dfrac{6(147)}{40} = \dfrac{441}{20} \, \text{°}\text{C} \nonumber \].
Exercise \(\PageIndex{6}\)
Find the average value of the function \(f(x,y,z) = xyz\) over the cube with sides of length 4 units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.
- Hint
-
Follow the steps in the previous example.
- Answer
-
\(f_{ave} = 8\)
Key Concepts
- To compute a triple integral we use Fubini’s theorem, which states that if \(f(x,y,z)\) is continuous on a rectangular box \(B = [a,b] \times [c,d] \times [e,f]\), then \[\iiint_B f(x,y,z) \,dV = \int_e^f \int_c^d \int_a^b f(x,y,z) \, dx \, dy \, dz \nonumber \] and is also equal to any of the other five possible orderings for the iterated triple integral.
- To compute the volume of a general solid bounded region \(E\) we use the triple integral \[V \, (E) = \iiint_E 1 \,dV. \nonumber \]
- Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
- To compute the average value of a function over a general three-dimensional region, we use \[f_{ave} = \dfrac{1}{V \, (E)} \iiint_E f(x,y,z) \,dV. \nonumber \]
Key Equations
- Triple integral
\[\lim_{l,m,n \rightarrow \infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \,\Delta x \Delta y \Delta z = \iiint_B f(x,y,z) \,dV \nonumber \]
Glossary
- triple integral
- the triple integral of a continuous function \(f(x,y,z)\) over a rectangular solid box \(B\) is the limit of a Riemann sum for a function of three variables, if this limit exists
Triple Integrals over a General Bounded Region
We now expand the definition of the triple integral to compute a triple integral over a more general bounded region \(E\) in \(\mathbb{R}^3\). The general bounded regions we will consider are of three types. First, let \(D\) be the bounded region that is a projection of \(E\) onto the \(xy\)-plane. Suppose the region \(E\) in \(\mathbb{R}^3\) has the form
\[E = \big\{(x,y,z)\,|\,(x,y) \in D, u_1(x,y) \leq z \leq u_2(x,y) \big\}. \nonumber \]
For two functions \(z = u_1(x,y)\) and \(u_2(x,y)\), such that \(u_1(x,y) \leq u_2(x,y)\) for all \((x,y)\) in \(D\) as shown in the following figure.


4.2: Grammar
In this section
- You will learn the difference between phrases and sentences.
- You will practice comparative and superlative forms.
- You will know how to use adverbs in different contexts.
- You will learn how to form the future tense and talk about future plans.
- You will create complex sentences using the structure of “because” and “because of.”
Grammar (1)
Definite and indefinite structures
Study the following examples to distinguish between phrases and sentences.
| The winter is cold | الشتاء بارد | This is a cold winter | هذا شتاء بارد |
| I do not like the cold weather | لا أحب الجو البارد | The weather is cold in winter | الجو بارد |
| The new restaurant is excellent | المطعم الجديد ممتاز | This is a new restaurant | هذا مطعم جديد |
Activity (4). Choose the right translation for each of the following sentences.
Grammar (2)
Comparatives and Superlatives
Study the following examples:
| The weather in Chicago is colder than the weather in Texas | الجو في شيكاغو أبرد من الجو في تكساس | ||
| New York city is bigger that Detroit city | مدينة نيويورك أكبر من مدينة ديترويت | ||
| The weather in the Fall is nicer than the weather in the Spring | الطقس في الخريف ألطف من الطقس في الربيع | ||
| The Arabic language book is easier than the history book | كتاب اللغة العربية أسهل من كتاب التاريخ | ||
| My father’s car is smaller than my mother’s car | سيارة والدي أصغر من سيارة والدتي | ||
| Sultan restaurant is the best restaurant in the city | مطعم (سلطان) هو أحسن مطعم في المدينة | ||
| What is the biggest state in America? | ما هي أكبر ولاية في أمريكا؟ |
The comparative form in Arabic is derived from the adjective in a predictable pattern as you see in the previous examples. In comparison, you should use the proposition من after the adjective while in the superlative form, you will use the identifier noun like in أحسن مطعم “best restaurant” and أكبر ولاية “biggest state”.
Study the following table on comparative and superlative forms:
[table id=40 /]
With your classmates. Create comparative sentences using the following prompts.

في رأيك …؟ بالنسبة لك …؟
(1) ما هي أحسن جامعة في العالم؟
(2) ما هي أبرد ولاية في أمريكا؟
(3) ما هو أحسن فيلم هذه السنة؟
(4) ما هو أصعب يوم في الأسبوع؟
(5) ما هو أحسن مطعم في المدينة؟
(6) ما هي أجمل مدينة في أمريكا؟
(7) من أحسن ممثل/ ممثلة actor في هوليوود؟
Grammar (3)
Adverbs
Study the following list of adverbs
[table id=37 /]
Grammar notes
The adverb in Arabic is usually derived from the adjective by adding a Tanween in the end. They describe the degree, frequency, or manner of the verb. Note the difference between these sentences.
|
خالد يذاكر
جيّداً
Khalid studies well |
خالد طالب
جيّد
Khalid is a good student |
|
آكل الدجاج
كثيراً
I eat chicken a lot |
آكل دجاج
كثير
I eat a lot of chicken |
|
هل تسكن
قريباً
؟
Do you live nearby? |
هل تسكن في بيت
قريب
؟
Do you live in a nearby house? |
- You note that the adjective describes the noun while the adverb describes the verb.
- Some adverbs describe adjectives as well like جداً “very” or أيضاً “also”.
- Adverbs usually come after adjectives or verbs but some of them can freely move around the sentence like دائماً “always,” أحياناً “sometimes,” and عادةً “usually”.
- The following expressions are equivalent:
| كُل يوم | يومياً |
| كُلّ أسبوع | أسبوعياً |
| في الصيف | صيفاً |
| في النهار | نهاراً |
| في الليل | ليلاً |
Choose the appropriate adjective that best fits in each of the following sentences
Find someone who. Survey your classmates to know if they do these activities. Report your findings to class in complete sentences.
| (1) | Travels in the summer? Where? | هل تسافر/ تسافرين صيفاً؟ أين؟ |
| (2) | Works at daytime? | …………………………………………………………… |
| (3) | Watches TV sometimes? | …………………………………………………………… |
| (4) | Studies well at the library? | …………………………………………………………… |
| (5) | Reads at night? | …………………………………………………………… |
| (6) | Lives nearby? | …………………………………………………………… |
| (7) | Goes to classes daily? | …………………………………………………………… |
| (8) | Eats chicken a lot | …………………………………………………………… |
| (9) | Very tired? | …………………………………………………………… |
| (10) | Always busy? | …………………………………………………………… |
| (11) | Speaks a lot with friends? | …………………………………………………………… |
| (12) | Sleeps late? | …………………………………………………………… |
Grammar (4)
Future Tense
Future tense in Arabic is formulated by adding the prefix ســ or the separate word سوف in front of a present tense verb. They can be used interchangeably. The use of سوف can imply a further distant and unplanned future while the use of سـ can be more immediate future. The negation of the future is formed by adding the word لَن before the present tense while removing the future marker.
- Common words and phrases that mark the future tense include:
| tomorrow | غداً |
| the day after tomorrow | بعد غد |
| the coming week | الأسبوع القادم |
| the coming month | الشهر القادم |
| the coming year | السنة القادمة |
| the coming summer | الصيف القادم |
| God willing | إن شاء الله |
Practice the verb “will travel” and its negation form:
[table id=41 /]
Fill in the blanks with the appropriate future verb using the root clues provided between brackets.
Ask your classmates about their future plans then report your findings to class. You can use the following prompts.
[table id=42 /]
Grammar (5)
“Because/Because of” Sentences
To express reason, you can use either لأنّ or بِسبب. These are used differently as لأنّ “because” should be followed by a complete sentence while بِسبب “because of” is followed by a noun or a nominal phrase but should not be a full sentence. Study the differences between these pairs of sentences:
|
لا أحب الشتاء لأنّ الجو بارد I do not like winter because the weather is cold |
لا أحب الشتاء بسبب الجو البارد I do not like winter because of the cold weather |
(1) |
|
أذهب إلى الجامعة يومياً لأنّ عندي فصول I go to university daily because I have classes |
أذهب إلى الجامعة يومياً بسبب الفصول I go to university daily because of classes |
(2) |
|
أختي تسكن في شيكاغو لأنّها تدرس في الجامعة My sister lives in Chicago because she studies in the university |
أختي تسكن في شيكاغو بسبب الدراسة My sister lives in Chicago because of study |
(3) |
Grammar notes
Note that لأنّ merges with the following pronoun, in which case is an object pronoun. Study the following table:
| لأنّها | لأن + هي | لأنّي | لأن + أنا |
| لأنّنا | لأن + نحن | لأنّكَ | لأن + أنتَ |
| لأنّكم | لأن + أنتم | لأنّكِ | لأن + أنتِ |
| لأنّهم | لأن + هم | لأنّه | لأن + هو |
Use بسبب or لأنّ to best complete the sentences below.
Write complete sentences using لأنّ or بسبب guided by picture and vocabulary clues.
1- أصحابي – مكتبة – امتحان
2-طلاب – فصل – كورونا
3-السكن – آلاسكا – الثلج
4-كلية الهندسة – يعمل – مهندس
5-السفر – فلوريدا – ديزني
Ask your classmates questions in Arabic using لماذا and report your findings to class. You can use the following prompts.
- Why do you study Arabic?
- Where do you like to travel? Why?
- In what area do you live? Do you like it? Why?
- Where do you like to work? Why?
- What is the best season in your opinion? Why?
- Where do you want to live in the future? Why?
Key Takeaways
- You can compare and contrast and express preferences.
- You can express reason or purpose in complex sentences.


15.16B: Genital Warts
- List the causes and various management procedures for genital warts
Genital warts (or Condylomata acuminata, venereal warts, anal warts and anogenital warts) is a highly contagious sexually transmitted disease caused by some sub-types of human papillomavirus (HPV ).   Warts are the most easily recognized symptom of genital HPV infection, where types 6 and 11 are responsible for 90% of genital warts cases.
Although it is estimated that only a “small percentage” (between 1% and 5%) of those infected with genital HPV develop genital warts, those infected can still transmit the virus. Other types of HPV also cause cervical cancer and probably most anal cancers, however it is important to underline that the types of HPV that cause the overwhelming majority of genital warts are not the same as those that can potentially increase the risk of genital or anal cancer. HPV prevalence at any one time has been observed in some studies at 27% over all sexually active people, rising to 45% between the ages of 14 and 19.
Diagnosis
Genital warts, histopathologically, characteristically rise above the skin surface due to enlargement of the dermal papillae, have parakeratosis and the characteristic nuclear changes typical of HPV infections (nuclear enlargement with perinuclear clearing).
Prevention
Gardasil (sold by Merck & Co.) is a vaccine that protects against human papillomavirus types 16, 18, 6, and 11. Types 6 and 11 cause genital warts, while 16 and 18 cause cervical cancer. The vaccine is preventive, not therapeutic, and must be given before exposure to the virus type to be effective, ideally before the beginning of sexual activity. The vaccine is widely approved for use by young women, it is being tested for young men, and has been approved for males in some areas, such as the UK, the US and Canada.
Management
There is no cure for HPV, but there are methods to treat visible warts, which could reduce infectivity, although there are no trials studying the effectiveness of removing visible warts in reducing transmission. Every year, Americans spend $200 million on the treatment of genital warts. Genital warts may disappear without treatment, but sometimes eventually develop a fleshy, small raised growth. There is no way to predict whether they will grow or disappear.
Warts can sometimes be identified because they show up as white when acetic acid is applied, but this method is not recommended on the vulva because microtrauma and inflammation can also show up as acetowhite. Magnifying glasses or colposcope may also be used to aid in identifying small warts. Depending on the sizes and locations of warts (as well as other factors), a doctor will offer one of several ways to treat them. Podofilox is the first-line treatment due to its low cost.Almost all treatments can potentially cause depigmentation or scarring. A 0.15% – 0.5% podophyllotoxin (also called podofilox) solution in a gel or cream. Marketed as Condylox (0.5%), Wartec (0.15%) and Warticon (0.15%), it can be applied by the patient to the affected area and is not washed off. It is the purified and standardized active ingredient of the podophyllin (see below). Podofilox is safer and more effective than podophyllin. Skin erosion and pain are more commonly reported than with imiquimod and sinecatechins. Its use is cycled (2 times per day for 3 days then 4–7 days off); one review states that it should only be used for four cycles.
Imiquimod (Aldara) is a topical immune response cream, applied to the affected area. It causes less local irritation than podofilox but may cause fungal infections (11% in package insert) and flu-like symptoms (less than 5% disclosed in package insert). Sinecatechins (marketed as Veregen and Polyphenon E) is an ointment of catechins (55% epigallocatechin gallate) extracted from green tea and other components. Mode of action is undetermined. It appears to have higher clearance rates than podophyllotoxin and imiquimod and causes less local irritation, but clearance takes longer than with imiquimod. Liquid nitrogen cryosurgery is safe for pregnancy. It kills warts 71–79% of the time, but recurrence is 38% to 73% 6 months after treatment. Local infections have been reported. Trichloroacetic acid (TCA) is less effective than cryosurgery, and is not recommended for use in the vagina, cervix, or urinary meatus. Surgical excision is best for large warts, and has a greater risk of scarring. Laser ablation does not seem to be any more effective than other physician-applied methods, but is often used as a last resort and is extremely expensive. A 20% podophyllin anti-mitotic solution, applied to the affected area and later washed off. However, this crude herbal extract is not recommended for use on vagina, urethra, perianal area, or cervix, and must be applied by a physician.
Reported reactions include nausea, vomiting, fever, confusion, coma, renal failure, ileus, and leukopenia; death has been reported with extensive topical application, or application on mucous membranes. Interferon can be used; it is effective, but it is also expensive and its effect is inconsistent. Electrocauterization can be used; it is an older procedure but recovery time is generally longer. In severe cases of genital warts, treatment may require general or spinal anesthesia. This is a surgical procedure. More effective than cryosurgery and recurrence is at a much lower rate. Oral Isotretinoin is a therapy that has proven effective in experimental use, but is rarely used due to potentially severe side effects. In a small-scale study, low dose oral isotretinoin showed considerable efficacy and may represent an alternative systemic form of therapy for Genital Warts. Yet, albeit this indicative evidence not many studies have been conducted to further confirm the findings. In most countries this therapy is currently unapproved and only used as an alternative therapy if other therapies failed.
Key Points
- Warts are the most easily recognized symptom of genital HPV infection, where types 6 and 11 are responsible for 90% of genital warts cases.

- Genital warts, histopathologically, characteristically rise above the skin surface due to enlargement of the dermal papillae, have parakeratosis and the characteristic nuclear changes typical of HPV infections (nuclear enlargement with perinuclear clearing).
- There is no cure for HPV, but there are methods to treat visible warts, which could reduce infectivity, although there are no trials studying the effectiveness of removing visible warts in reducing transmission.
Key Terms
- human papillomavirus : A virus that affects humans, sometimes causing cervical or other cancer; it is sometimes classified as a sexually transmitted disease.


3.6: How to Read Like a Writer
by Mike Bunn
In 1997, I was a recent college graduate living in London for six months and working at the Palace Theatre owned by Andrew Lloyd Webber. The Palace was a beautiful red brick, four-story theatre in the heart of London’s famous West End, and eight times a week it housed a three-hour performance of the musical Les Miserables . Because of antiquated fire-safety laws, every theatre in the city was required to have a certain number of staff members inside watching the performance in case of an emergency.
Figure \(\PageIndex{1}\)
My job (in addition to wearing a red tuxedo jacket) was to sit inside the dark theater with the patrons and make sure nothing went wrong. It didn’t seem to matter to my supervisor that I had no training in security and no idea where we kept the fire extinguishers. I was pretty sure that if there was any trouble I’d be running down the back stairs, leaving the patrons to fend for themselves. I had no intention of dying in a bright red tuxedo.
There was a Red Coat stationed on each of the theater’s four floors, and we all passed the time by sitting quietly in the back, reading books with tiny flashlights. It’s not easy trying to read in the dim light of a theatre—flashlight or no flashlight—and it’s even tougher with shrieks and shouts and gunshots coming from the stage. I had to focus intently on each and every word, often rereading a single sentence several times. Sometimes I got distracted and had to re-read entire paragraphs. As I struggled to read in this environment, I began to realize that the way I was reading—one word at a time—was exactly the same way that the author had written the text. I realized writing is a word-by-word, sentence-by-sentence process. The intense concentration required to read in the theater helped me recognize some of the interesting ways that authors string words into phrases into paragraphs into entire books.
I came to realize that all writing consists of a series of choices.
I was an English major in college, but I don’t think I ever thought much about reading. I read all the time. I read for my classes and on the computer and sometimes for fun, but I never really thought about the important connections between reading and writing, and how reading in a particular way could also make me a better writer.
What Does It Mean to Read Like a Writer?
When you Read Like a Writer (RLW) you work to identify some of the choices the author made so that you can better understand how such choices might arise in your own writing. The idea is to carefully examine the things you read, looking at the writerly techniques in the text in order to decide if you might want to adopt similar (or the same) techniques in your writing.
You are reading to learn about writing.
Instead of reading for content or to better understand the ideas in the writing (which you will automatically do to some degree anyway), you are trying to understand how the piece of writing was put together by the author and what you can learn about writing by reading a particular text. As you read in this way, you think about how the choices the author made and the techniques that he/she used are influencing your own responses as a reader. What is it about the way this text is written that makes you feel and respond the way you do?
Figure \(\PageIndex{2}\)
The goal as you read like a writer is to locate what you believe are the most important writerly choices represented in the text—choices as large as the overall structure or as small as a single word used only once—to consider the effect of those choices on potential readers (including yourself). Then you can go one step further and imagine what different choices the author might have made instead, and what effect those different choices would have on readers.
Say you’re reading an essay in class that begins with a short quote from President Barack Obama about the war in Iraq. As a writer, what do you think of this technique? Do you think it is effective to begin the essay with a quote? What if the essay began with a quote from someone else? What if it was a much longer quote from President Obama, or a quote from the President about something other than the war?
And here is where we get to the most important part: Would you want to try this technique in your own writing?
Would you want to start your own essay with a quote? Do you think it would be effective to begin your essay with a quote from President Obama? What about a quote from someone else? You could make yourself a list. What are the advantages and disadvantages of starting with a quote? What about the advantages and disadvantages of starting with a quote from the President? How would other readers respond to this technique? Would certain readers (say Democrats or liberals) appreciate an essay that started with a quote from President Obama better than other readers (say Republicans or conservatives)? What would be the advantages and disadvantages of starting with a quote from a less divisive person? What about starting with a quote from someone more divisive?
The goal is to carefully consider the choices the author made and the techniques that he or she used, and then decide whether you want to make those same choices or use those same techniques in your own writing. Author and professor Wendy Bishop explains how her reading process changed when she began to read like a writer:
It wasn’t until I claimed the sentence as my area of desire, interest, and expertise—until I wanted to be a writer writing better—that I had to look underneath my initial readings . . . I started asking, how — how did the writer get me to feel, how did the writer say something so that it remains in my memory when many other things too easily fall out, how did the writer communicate his/her intentions about genre, about irony? (119–20)
Bishop moved from simply reporting her personal reactions to the things she read to attempting to uncover how the author led her (and other readers) to have those reactions. This effort to uncover how authors build texts is what makes Reading Like a Writer so useful for student writers.
How Is RLW Different from “Normal” Reading?
Most of the time we read for information. We read a recipe to learn how to bake lasagna. We read the sports page to see if our school won the game, Facebook to see who has commented on our status update, a history book to learn about the Vietnam War, and the syllabus to see when the next writing assignment is due. Reading Like a Writer asks for something very different.
In 1940, a famous poet and critic named Allen Tate discussed two different ways of reading:
There are many ways to read, but generally speaking there are two ways. They correspond to the two ways in which we may be interested in a piece of architecture. If the building has Corinthian columns, we can trace the origin and development of Corinthian columns; we are interested as historians. But if we are interested as architects, we may or may not know about the history of the Corinthian style; we must, however, know all about the construction of the building, down to the last nail or peg in the beams. We have got to know this if we are going to put up buildings ourselves. (506)
While I don’t know anything about Corinthian columns (and doubt that I will ever want to know anything about Corinthian columns), Allen Tate’s metaphor of reading as if you were an architect is a great way to think about RLW. When you read like a writer, you are trying to figure out how the text you are reading was constructed so that you learn how to “build” one for yourself. Author David Jauss makes a similar comparison when he writes that “reading won’t help you much unless you learn to read like a writer. You must look at a book the way a carpenter looks at a house someone else built, examining the details in order to see how it was made” (64).
Perhaps I should change the name and call this Reading Like an Architect, or Reading Like a Carpenter. In a way those names make perfect sense. You are reading to see how something was constructed so that you can construct something similar yourself.
Why Learn to Read Like a Writer?
For most college students RLW is a new way to read, and it can be difficult to learn at first. Making things even more difficult is that your college writing instructor may expect you to read this way for class but never actually teach you how to do it. He or she may not even tell you that you’re supposed to read this way. This is because most writing instructors are so focused on teaching writing that they forget to show students how they want them to read.
That’s what this essay is for.
Figure \(\PageIndex{3}\)
In addition to the fact that your college writing instructor may expect you to read like a writer, this kind of reading is also one of the very best ways to learn how to write well. Reading like a writer can help you understand how the process of writing is a series of making choices, and in doing so, can help you recognize important decisions you might face and techniques you might want to use when working on your own writing. Reading this way becomes an opportunity to think and learn about writing.
Charles Moran, a professor of English at the University of Massachusetts, urges us to read like writers because:
When we read like writers we understand and participate in the writing. We see the choices the writer has made, and we see how the writer has coped with the consequences of those choices . . . We “see” what the writer is doing because we read as writers; we see because we have written ourselves and know the territory, know the feel of it, know some of the moves ourselves. (61)
You are already an author, and that means you have a built-in advantage when reading like a writer. All of your previous writing experiences—inside the classroom and out—can contribute to your success with RLW. Because you “have written” things yourself, just as Moran suggests, you are better able to “see” the choices that the author is making in the texts that you read. This in turn helps you to think about whether you want to make some of those same choices in your own writing, and what the consequences might be for your readers if you do.
What Are Some Questions to Ask Before You Start Reading?
As I sat down to work on this essay, I contacted a few of my former students to ask what advice they would give to college students regarding how to read effectively in the writing classroom and also to get their thoughts on RLW. Throughout the rest of the essay I’d like to share some of their insights and suggestions; after all, who is better qualified to help you learn what you need to know about reading in college writing courses than students who recently took those courses themselves?
One of the things that several students mentioned to do first, before you even start reading, is to consider the context surrounding both the assignment and the text you’re reading. As one former student, Alison, states: “The reading I did in college asked me to go above and beyond, not only in breadth of subject matter, but in depth, with regards to informed analysis and background information on context .” Alison was asked to think about some of the factors that went into the creation of the text, as well as some of the factors influencing her own experience of reading—taken together these constitute the context of reading. Another former student, Jamie, suggests that students “learn about the historical context of the writings” they will read for class. Writing professor Richard Straub puts it this way: “You’re not going to just read a text. You’re going to read a text within a certain context, a set of circumstances . . . It’s one kind of writing or another, designed for one audience and purpose or another” (138).
Among the contextual factors you’ll want to consider before you even start reading are:
- Do you know the author’s purpose for this piece of writing?
- Do you know who the intended audience is for this piece of writing?
It may be that you need to start reading before you can answer these first two questions, but it’s worth trying to answer them before you start. For example, if you know at the outset that the author is trying to reach a very specific group of readers, then his or her writerly techniques may seem more or less effective than if he/she was trying to reach a more general audience. Similarly—returning to our earlier example of beginning an essay with a quote from President Obama about the war in Iraq—if you know that the author’s purpose is to address some of the dangers and drawbacks of warfare, this may be a very effective opening. If the purpose is to encourage Americans to wear sunscreen while at the beach this opening makes no sense at all. One former student, Lola, explained that most of her reading assignments in college writing classes were designed “to provoke analysis and criticisms into the style, structure, and purpose of the writing itself.”
In What Genre Is This Written?
Figure \(\PageIndex{4}\)
Another important thing to consider before reading is the genre of the text. Genre means a few different things in college English classes, but it’s most often used to indicate the type of writing: a poem, a newspaper article, an essay, a short story, a novel, a legal brief, an instruction manual, etc. Because the conventions for each genre can be very different (who ever heard of a 900-page newspaper article?), techniques that are effective for one genre may not work well in another. Many readers expect poems and pop songs to rhyme, for example, but might react negatively to a legal brief or instruction manual that did so. Another former student, Mike, comments on how important the genre of the text can be for reading:
I think a lot of the way I read, of course, depends on the type of text I’m reading. If I’m reading philosophy, I always look for signaling words (however, therefore, furthermore, despite) indicating the direction of the argument . . . when I read fiction or creative nonfiction, I look for how the author inserts dialogue or character sketches within narration or environmental observation. After reading To the Lighthouse [sic] last semester, I have noticed how much more attentive I’ve become to the types of narration (omniscient, impersonal, psychological, realistic, etc.), and how these different approaches are utilized to achieve an author’s overall effect.
Although Mike specifically mentions what he looked for while reading a published novel, one of the great things about RLW is that it can be used equally well with either published or student-produced writing.
Is This a Published or a Student-Produced Piece of Writing?
As you read both kinds of texts you can locate the choices the author made and imagine the different decisions that he/she might have made. While it might seem a little weird at first to imagine how published texts could be written differently—after all, they were good enough to be published—remember that all writing can be improved. Scholar Nancy Walker believes that it’s important for students to read published work using RLW because “the work ceases to be a mere artifact, a stone tablet, and becomes instead a living utterance with immediacy and texture. It could have been better or worse than it is had the author made different choices” (36). As Walker suggests, it’s worth thinking about how the published text would be different—maybe even better —if the author had made different choices in the writing because you may be faced with similar choices in your own work.
Is This the Kind of Writing You Will Be Assigned to Write Yourself?
Knowing ahead of time what kind of writing assignments you will be asked to complete can really help you to read like a writer. It’s probably impossible (and definitely too time consuming) to identify all of the choices the author made and all techniques an author used, so it’s important to prioritize while reading. Knowing what you’ll be writing yourself can help you prioritize. It may be the case that your instructor has assigned the text you’re reading to serve as model for the kind of writing you’ll be doing later. Jessie, a former student, writes, “In college writing classes, we knew we were reading for a purpose—to influence or inspire our own work. The reading that I have done in college writing courses has always been really specific to a certain type of writing, and it allows me to focus and experiment on that specific style in depth and without distraction.”
If the text you’re reading is a model of a particular style of writing—for example, highly-emotional or humorous—RLW is particularly helpful because you can look at a piece you’re reading and think about whether you want to adopt a similar style in your own writing. You might realize that the author is trying to arouse sympathy in readers and examine what techniques he/she uses to do this; then you can decide whether these techniques might work well in your own writing. You might notice that the author keeps including jokes or funny stories and think about whether you want to include them in your writing—what would the impact be on your potential readers?
What Are Questions to Ask As You Are Reading?
It is helpful to continue to ask yourself questions as you read like a writer. As you’re first learning to read in this new way, you may want to have a set of questions written or typed out in front of you that you can refer to while reading. Eventually—after plenty of practice—you will start to ask certain questions and locate certain things in the text almost automatically. Remember, for most students this is a new way of reading, and you’ll have to train yourself to do it well. Also keep in mind that you’re reading to understand how the text was written —how the house was built—more than you’re trying to determine the meaning of the things you read or assess whether the texts are good or bad.
First, return to two of the same questions I suggested that you consider before reading:
- What is the author’s purpose for this piece of writing?
- Who is the intended audience?
Figure \(\PageIndex{5}\)
Think about these two questions again as you read. It may be that you couldn’t really answer them before, or that your ideas will change while reading. Knowing why the piece was written and who it’s for can help explain why the author might have made certain choices or used particular techniques in the writing, and you can assess those choices and techniques based in part on how effective they are in fulfilling that purpose and/or reaching the intended audience.
Beyond these initial two questions, there is an almost endless list of questions you might ask regarding writing choices and techniques. Here are some of the questions that one former student, Clare, asks herself:
When reading I tend to be asking myself a million questions. If I were writing this, where would I go with the story? If the author goes in a different direction (as they so often do) from what I am thinking, I will ask myself, why did they do this? What are they telling me?
Clare tries to figure out why the author might have made a move in the writing that she hadn’t anticipated, but even more importantly, she asks herself what she would do if she were the author. Reading the text becomes an opportunity for Clare to think about her own role as an author.
Here are some additional examples of the kinds of questions you might ask yourself as you read:
- How effective is the language the author uses? Is it too formal? Too informal? Perfectly appropriate?
Depending on the subject matter and the intended audience, it may make sense to be more or less formal in terms of language. As you begin reading, you can ask yourself whether the word choice and tone/ language of the writing seem appropriate.
- What kinds of evidence does the author use to support his/her claims? Does he/she use statistics? Quotes from famous people? Personal anecdotes or personal stories? Does he/she cite books or articles?
- How appropriate or effective is this evidence? Would a different type of evidence, or some combination of evidence, be more effective?
To some extent the kinds of questions you ask should be determined by the genre of writing you are reading. For example, it’s probably worth examining the evidence that the author uses to support his/ her claims if you’re reading an opinion column, but less important if you’re reading a short story. An opinion column is often intended to convince readers of something, so the kinds of evidence used are often very important. A short story may be intended to convince readers of something, sometimes, but probably not in the same way. A short story rarely includes claims or evidence in the way that we usually think about them.
- Are there places in the writing that you find confusing? What about the writing in those places makes it unclear or confusing?
It’s pretty normal to get confused in places while reading, especially while reading for class, so it can be helpful to look closely at the writing to try and get a sense of exactly what tripped you up. This way you can learn to avoid those same problems in your own writing.
- How does the author move from one idea to another in the writing? Are the transitions between the ideas effective? How else might he/she have transitioned between ideas instead?
Figure \(\PageIndex{6}\)
Notice that in these questions I am encouraging you to question whether aspects of the writing are appropriate and effective in addition to deciding whether you liked or disliked them. You want to imagine how other readers might respond to the writing and the techniques you’ve identified. Deciding whether you liked or disliked something is only about you; considering whether a technique is appropriate or effective lets you contemplate what the author might have been trying to do and to decide whether a majority of readers would find the move successful. This is important because it’s the same thing you should be thinking about while you are writing: how will readers respond to this technique I am using, to this sentence, to this word? As you read, ask yourself what the author is doing at each step of the way, and then consider whether the same choice or technique might work in your own writing.
What Should You Be Writing As You Are Reading?
The most common suggestion made by former students—mentioned by every single one of them—was to mark up the text, make comments in the margins, and write yourself notes and summaries both during and after reading. Often the notes students took while reading became ideas or material for the students to use in their own papers. It’s important to read with a pen or highlighter in your hand so that you can mark—right on the text—all those spots where you identify an interesting choice the author has made or a writerly technique you might want to use. One thing that I like to do is to highlight and underline the passage in the text itself, and then try to answer the following three questions on my notepad:
- What is the technique the author is using here?
- Is this technique effective?
- What would be the advantages and disadvantages if I tried this same technique in my writing?
By utilizing this same process of highlighting and note taking, you’ll end up with a useful list of specific techniques to have at your disposal when it comes time to begin your own writing.
What Does RLW Look Like in Action?
Let’s go back to the opening paragraph of this essay and spend some time reading like writers as a way to get more comfortable with the process:
In 1997, I was a recent college graduate living in London for six months and working at the Palace Theatre owned by Andrew Lloyd Webber. The Palace was a beautiful red brick, four-story theatre in the heart of London’s famous West End, and eight times a week it housed a three-hour performance of the musical Les Miserables . Because of antiquated fire-safety laws, every theatre in the city was required to have a certain number of staff members inside watching the performance in case of an emergency.
Let’s begin with those questions I encouraged you to try to answer before you start reading. (I realize we’re cheating a little bit in this case since you’ve already read most of this essay, but this is just practice. When doing this on your own, you should attempt to answer these questions before reading, and then return to them as you read to further develop your answers.)
- Do you know the author’s purpose for this piece of writing? I hope the purpose is clear by now; if it isn’t, I’m doing a pretty lousy job of explaining how and why you might read like a writer.
- Do you know who the intended audience is? Again, I hope that you know this one by now.
- What about the genre? Is this an essay? An article? What would you call it?
- You know that it’s published and not student writing. How does this influence your expectations for what you will read?
- Are you going to be asked to write something like this yourself? Probably not in your college writing class, but you can still use RLW to learn about writerly techniques that you might want to use in whatever you do end up writing.
Now ask yourself questions as you read.
In 1997, I was a recent college graduate living in London for six months and working at the Palace Theatre owned by Andrew Lloyd Webber. The Palace was a beautiful red brick, four-story theatre in the heart of London’s famous West End, and eight times a week it housed a three-hour performance of the musical Les Miserables . Because of antiquated fire-safety laws, every theatre in the city was required to have a certain number of staff members inside watching the performance in case of an emergency.
Figure \(\PageIndex{7}\)
Since this paragraph is the very first one, it makes sense to think about how it introduces readers to the essay. What technique(s) does the author use to begin the text? This is a personal story about his time working in London. What else do you notice as you read over this passage? Is the passage vague or specific about where he worked? You know that the author worked in a famous part of London in a beautiful theater owned by a well-known composer. Are these details important? How different would this opening be if instead I had written:
In 1997, I was living in London and working at a theatre that showed Les Miserables .
This is certainly shorter, and some of you may prefer this version. It’s quick. To the point. But what (if anything) is lost by eliminating so much of the detail? I chose to include each of the details that the revised sentence omits, so it’s worth considering why. Why did I mention where the theater was located? Why did I explain that I was living in London right after finishing college? Does it matter that it was after college? What effect might I have hoped the inclusion of these details would have on readers? Is this reference to college an attempt to connect with my audience of college students? Am I trying to establish my credibility as an author by announcing that I went to college? Why might I want the readers to know that this was a theater owned by Andrew Lloyd Weber? Do you think I am just trying to mention a famous name that readers will recognize? Will Andrew Lloyd Weber figure prominently in the rest of the essay?
These are all reasonable questions to ask. They are not necessarily the right questions to ask because there are no right questions. They certainly aren’t the only questions you could ask, either. The goal is to train yourself to formulate questions as you read based on whatever you notice in the text. Your own reactions to what you’re reading will help determine the kinds of questions to ask.
Now take a broader perspective. I begin this essay—an essay about reading —by talking about my job in a theater in London. Why? Doesn’t this seem like an odd way to begin an essay about reading? If you read on a little further (feel free to scan back up at the top of this essay) you learn in the third full paragraph what the connection is between working in the theater and reading like a writer, but why include this information at all? What does this story add to the essay? Is it worth the space it takes up?
Think about what effect presenting this personal information might have on readers. Does it make it feel like a real person, some “ordinary guy,” is talking to you? Does it draw you into the essay and make you want to keep reading?
What about the language I use? Is it formal or more informal? This is a time when you can really narrow your focus and look at particular words:
Because of antiquated fire-safety laws, every theatre in the city was required to have a certain number of staff members inside watching the performance in case of an emergency.
 It certainly projects a negative impression; if the laws are described as antiquated it means I view them as old-fashioned or obsolete. This is a fairly uncommon word, so it stands out, drawing attention to my choice in using it. The word also sounds quite formal. Am I formal in the rest of this sentence?
I use the word “performance” when I just as easily could have written “show.” For that matter, I could have written “old” instead of “antiquated.” You can proceed like this throughout the sentence, thinking about alternative choices I could have made and what the effect would be. Instead of “staff members” I could have written “employees” or just “workers.” Notice the difference if the sentence had been written:
Because of old fire-safety laws, every theatre in the city was required to have a certain number of workers inside watching the show in case of an emergency.
Which version is more likely to appeal to readers? You can try to answer this question by thinking about the advantages and disadvantages of using formal language. When would you want to use formal language in your writing and when would it make more sense to be more conversational?
As you can see from discussing just this one paragraph, you could ask questions about the text forever. Luckily, you don’t have to. As you continue reading like a writer, you’ll learn to notice techniques that seem new and pay less attention to the ones you’ve thought about before. The more you practice the quicker the process becomes until you’re reading like a writer almost automatically.
I want to end this essay by sharing one more set of comments by my former student, Lola, this time about what it means to her to read like a writer:
Reading as a writer would compel me to question what might have brought the author to make these decisions, and then decide what worked and what didn’t. What could have made that chapter better or easier to understand? How can I make sure I include some of the good attributes of this writing style into my own? How can I take aspects that I feel the writer failed at and make sure not to make the same mistakes in my writing?
Questioning why the author made certain decisions. Considering what techniques could have made the text better. Deciding how to include the best attributes of what you read in your own writing. This is what Reading Like a Writer is all about.
Are you ready to start reading?
Discussion
- How is “Reading Like a Writer” similar to and/or different from the way(s) you read for other classes?
- What kinds of choices do you make as a writer that readers might identify in your written work?
- Is there anything you notice in this essay that you might like to try in your own writing? What is that technique or strategy? When do you plan to try using it?
- What are some of the different ways that you can learn about the context of a text before you begin reading it?
Works Cited
Bishop, Wendy. “Reading, Stealing, and Writing Like a Writer.” Elements of Alternate Style: Essays on Writing and Revision. Ed. Wendy Bishop. Portsmouth, NH: Boynton/Cook, 1997. Print.
Jauss, David. “Articles of Faith.” Creative Writing in America: Theory and Pedagogy. Ed. Joseph Moxley. Urbana, IL: NCTE, 1989. Print.
Moran, Charles. “Reading Like a Writer.” Vital Signs 1. Ed. James L. Collins. Portsmouth, NH: Boynton/Cook, 1990. Print.
Straub, Richard. “Responding—Really Responding—to Other Students’ Writing.” The Subject is Reading. Ed. Wendy Bishop. Portsmouth, NH: Boynton/Cook, 2000. Print.
Tate, Allen. “We Read as Writers.” Princeton Alumni Weekly 40 (March 8, 1940): 505- 506. Print.
Walker, Nancy. “The Student Reader as Writer.” ADE Bulletin 106 (1993) 35–37. Print.


Chapter 8: Thermal Physics Last updated Save as PDF Page ID 46198 OpenStax OpenStax 8.1: Introduction to Thermal Physics 8.2: Temperature 8.3: The Ideal Gas Law 8.4: Heat 8.5: Heat Transfer Methods 8.6: Temperature Change and Heat Capacity 8.7: Phase Change and Latent Heat 8.8: The First Law of Thermodynamics 8.9: The First Law of Thermodynamics and Heat Engine Processes 8.10: Introduction to the Second Law of Thermodynamics- Heat Engines and Their Efficiency 8.11: Carnot’s Perfect Heat Engine- The Second Law of Thermodynamics Restated 8.12: Applications of Thermodynamics- Heat Pumps and Refrigerators 8.13: Entropy and the Second Law of Thermodynamics- Disorder and the Unavailability of Energy 8.14: Statistical Interpretation of Entropy and the Second Law of Thermodynamics- The Underlying Explanation 8.E: Thermal Physics (Exercises)


19.2D: Lymphatic Capillaries
Lymph capillaries are tiny, thin-walled vessels, closed at one end and located in the spaces between cells throughout the body.
- Describe the location, structure, and role of lymphatic capillaries in maintaining the pressure of the interstitial fluid
Key Points
- Lymph or lymphatic capillaries are tiny thin-walled vessels, closed at one end and located in the spaces between cells throughout the body, except in the central nervous system and non-vascular tissues.
- Lymphatic capillaries are slightly larger in diameter and have greater oncotic pressure than blood capillaries.
- When pressure is greater in the interstitial fluid than in lymph, the minivalve cells separate slightly and interstitial fluid enters the lymphatic capillary. When pressure is greater inside the lymphatic capillary, the cells of the minivalves adhere more closely, and lymph cannot flow back into interstitial fluid.
- Anchoring filaments attach to the minivalves to anchor the capillary to connective tissue, and also pull the capillary open to increase lymph collection when the tissue is swollen.
- Because lymph capillaries have a closed end, lymph is pushed forward into larger vessels as the pressure inside the capillary increases as lymph accumulates from fluid collection.
- Edema can occur when interstitial fluid accumulation in tissues is greater than fluid removal (acute inflammation ) or when the lymph vessels are obstructed in some way (elephantiasis).
Key Terms
- interstitial fluid : Also called tissue fluid, a solution that bathes and surrounds the cells of multicellular animals.
- lymph capillaries : Tiny thin-walled vessels, closed at one end and located in the spaces between cells throughout the body, collect fluid from the tissues.
Lymphatic circulation begins in the smallest type of lymph vessels, the lymph capillaries. These regulate the pressure of interstitial fluid by draining lymph from the tissues.
Structure of Lymphatic Capillaries
Lymph or lymphatic capillaries are tiny thin-walled vessels, closed at one end and located in the spaces between cells throughout the body. These are particularly dense within connective tissue. Lymphatic capillaries are slightly larger in diameter than blood capillaries and contain flap-like “minivalves” that permit interstitial fluid to flow into them but not out, under normal conditions.
Lymphatic capillaries are primarily made out of an endothelium layer that sits on a permeable basement membrane. The flap-like minivalves, located at gap-like junctions in the endothelium, are formed from the overlap of endothelial cells and are normally closed. Attached to the outer opening of the minivalves are anchoring filaments containing elastic fibers. They extend out from the lymphatic capillary, attaching the endothelium to fibroblast cells in the connective tissue. Unlike larger lymphatic vessels, lymphatic capillaries do not contain smooth muscle nor do they have a well developed adventitia, only small elastic filaments that perform a similar function.
Function of Lymphatic Capillaries
The lymph capillaries serve a variety of important functions.
Fluid Pressure Regulation
Lymphatic capillaries collect lymph fluid from the tissues, which allows them to regulate the pressure of interstitial fluid. This fluid is essentially plasma that leaks out of cardiovascular capillaries into the tissues due to the forces of hydrostatic or oncotic pressure. When pressure is greater in the interstitial fluid than in lymph due to accumulation of interstitial fluid, the minivalves separate slightly like the opening of a one-way swinging door so that fluid can enter the lymphatic capillary. When pressure is greater inside the lymphatic capillary, the cells adhere more closely to each other to prevent lymph backflow. The anchoring filaments are also pulled when the tissues are swollen. This opens the lymph capillaries more, increasing their volume and reducing their pressure to further facilitate fluid flow into the capillaries.
Lymph capillaries have a greater oncotic pressure (a pulling pressure exerted by proteins in solution) than blood plasma due to the greater concentration of plasma proteins in lymph. Additionally, the greater size of lymphatic capillaries compared to cardiovascular capillaries allows them to take more fluid proteins into lymph compared to plasma, which is the other reason for their greater levels of oncotic pressure. This also explains why lymph flows into the lymph capillaries easily, since fluid follows proteins that exert oncotic pressure.
Edema Prevention
Under normal conditions, lymph capillaries prevent the accumulation of edema (abnormal swelling) in the tissues. However, edema will still occur during acute inflammation or diseases in which lymph vessels are obstructed. During inflammation, fluid leaks into the tissues at a rate faster than it can be removed by the lymph capillaries due to the increased permeability of cardiovascular capillaries. During lymph vessel obstruction (such as through elephantiasis infection), lymph will be unable to progress normally through the lymphatic system, and pressure within the blocked off lymph capillaries increases to the point where backflow into tissues may occur, while the pressure of interstitial fluid gradually rises.
Drive Lymph Through Lymphatic Vessels
Lymph Capillary : Diagram showing the formation of lymph from interstitial fluid (labeled here as “tissue fluid”). Note: how the tissue fluid is entering the blind ends of lymph capillaries (indicated by deep green arrows).
The lymphatic capillaries bring lymph further into the lymphatic vessels. The capillaries have external valves but no internal valves or smooth muscle, so the pressure of lymph accumulation itself must propel the fluid forward into the larger vessels. Because lymphatic capillaries have a closed end and minivalves normally prevent backflow into tissues, the pressure of lymph becomes higher as more lymph is collected from the tissues, which sends the lymph fluid forward. Multiple capillaries converge in collecting vessels, where the internal valves and smooth muscle start to appear. This moves lymph further along the system despite the fall in pressure that occurs when moving from the higher-pressure capillaries to the lower-pressure collecting vessels.


3.5: Anita Desai (1937 - )
Born in British India, Anita Desai’s parents were Dhiren Mazumdar, an Indian businessman, and Antoinette Nim, a German. She spoke German and Hindi at home and learned English at school. Despite her parents’ anti-British sentiments, Desai attended Dehli’s Queen Mary’s Higher Secondary School, a school run by British Catholic nuns. She then attended Miranda House, a women’s college of Dehli University. She majored in English literature and published her first short story while still a student.
After she graduated in 1957, Desai worked at the German Cultural Institute where she met Ashuin Desai, whom she married. She moved with him to various places, including Mumbai and Chandigarh, before they settled in Prune. She also continued her writing, publishing her first novel, Cry, the Peacock, in 1963. She followed this with short story collections and novels addressing postcolonial issues in India and women’s experience in the domestic realm within the larger social and cultural context of middle class, urban India. Her novel In Custody (1984) was short-listed for the Booker Prize and made into a film.
Desai gained recognition as a writer of profound psychological insight on the displaced and the alienated who often face prejudice and who often suffer a fractured sense of identity. Her style is characterized by its precision and careful crafting. She has won such important literary prizes as the Indian National Academy of Letters Award and the Benson Medal from London’s Royal Society of Literature. She has taught at various prestigious universities around the world, and is professor emeritus at the Massachusetts Institute of Technology.


7.6: Antiparasitic and Anthelminthic Drugs
By the end of this section, you should be able to:
- 7.6.1 Describe the pathophysiology of common parasitic and helminthic infections.
- 7.6.2 Identify clinical manifestations related to common parasitic and helminthic infections.
- 7.6.3 Identify the etiology and diagnostic studies related to common parasitic and helminthic infections.
- 7.6.4 Identify the characteristics of drugs used to treat common parasitic and helminthic infections.
- 7.6.5 Explain the indications, actions, adverse reactions, and interactions of drugs used to treat common parasitic and helminthic infections.
- 7.6.6 Describe nursing implications of drugs used to treat common parasitic and helminthic infections.
- 7.6.7 Explain the client education related to drugs used to treat common parasitic and helminthic infections.
Parasitic Infections
In the United States, parasitic infections can affect anyone, but they disproportionately affect immunocompromised individuals, members of racial and ethnic minority groups, and those with low income. Parasitic infections are defined by the relationship been the host (human) and the parasite. Common types of parasitic infections are caused by protozoa and helminths.
- Protozoa : Protozoa are unicellular organisms that replicate within a human host. Examples of protozoal infections include those caused by Giardia, Plasmodium , and Babesia . Many of these conditions affect clients living in areas with poor sanitation because the spread can be attributed to either fecal-contaminated water or fecal–oral transfer. Signs and symptoms of protozoal infections include diarrhea, low-grade fever, nausea, greasy stools, and depressed appetite. Intestinal protozoal infections are most often diagnosed from microscopic stool samples, although molecular tests have recently been developed that may allow faster detection of infections.
- Helminths : Helminths, or worms, are large complex organisms that can be consumed while they are in a larval stage and then mature into their adult stage in the host’s gastrointestinal tract. Common signs and symptoms include abdominal pain and diarrhea; however, infections can also be asymptomatic. Similar to protozoal infections, microscopic examination of stool samples can reveal helminth eggs and aid in diagnosis.
Antiparasitic Drugs
This section covers the most frequently used antiparasitic drugs , including their mechanisms, adverse effects, indications, and contraindications. It is important to ensure that an accurate diagnosis has been made regarding the causative organism because this will dictate which medication is most appropriate for the client. Some of the most common antiparasitic drugs include:
- Metronidazole: For Giardia lamblia, Trichomonas vaginalis, Cryptosporidium parvum , and Toxoplasmosis gondii , metronidazole is the drug of choice. It provides good coverage against Entamoeba, Giardia , and Trichomonas , which are the species that cause infection.
- Tinidazole: Tinidazole is structurally related to metronidazole and has similar actions, adverse effects, and interactions. It also has a longer half-life than metronidazole, so dosing is more convenient. However, tinidazole is much more expensive than metronidazole. Tinidazole is contraindicated in clients consuming alcohol while using the drug due to risk for causing a severe flushing reaction.
- Nitazoxanide: Nitazoxanide is an alternative antiparasitic that can be used to treat giardiasis and cryptosporidiosis. Its full mechanism is unknown, but it is thought to disrupt the ability of these microorganisms to undergo anaerobic metabolism, leading to cell death. Nitazoxanide is well tolerated but can cause some gastrointestinal signs and symptoms.
- Scabicides and pediculicides: Scabicides and pediculicides are designed to treat infestations caused by lice and scabies. Lice, depending on the species, can cause symptoms on the head, body, or genitals. Collectively, this condition is known as pediculosis. Lice can be difficult to treat because if the eggs (nits) are not successfully removed from the hair after treatment, reinfestation can occur. Nit combs can be indispensable along with pediculicides. All agents in this category are applied topically to the affected area.
- Permethrin: Permethrin has a wide range of activity against a variety of arthropods, which makes it useful for lice infestations. Permethrin works by causing neuronal hyperpolarization and paralysis in the organism, leading to death.
- Lindane: Lindane has activity against lice and scabies and works by being absorbed into the parasite’s exoskeleton, leading to seizures and death.
- Malathion: Malathion comes from a group of chemicals known as organophosphates, which have been used as agricultural insecticides for many years. It works by inhibiting the enzyme acetylcholinesterase, which is responsible for metabolizing acetylcholine. This causes a buildup of acetylcholine, leading to overstimulation, paralysis, and death of the parasite.
- Spinosad: Spinosad is another topical antiparasitic with activity against scabies and lice. It works by inducing neuronal excitation and eventual paralysis and death of the organism. The most common adverse effects seen with spinosad include redness and skin irritation at the application site.
Clinical Tip
Ensure Removal of Nits
Nits are the eggs of lice that attach to the hair, and they are unaffected by pediculicides. To ensure that nits do not go on to mature into adults requiring further pediculicide treatment, it is important to carefully comb through the hair with a fine-tooth nit comb after using a pediculicide. Shaving the head is a viable alternative, but this is not a desirable outcome for many clients.
Anthelminthic Drugs
The available anthelminthic agents are broad spectrum and can treat a variety of worms, including roundworms, pinworms, hookworms, and whipworms. Because worm infections occur in the gastrointestinal tract, these agents are taken orally.
- Mebendazole: Mebendazole works by inhibiting microtubule formation in susceptible helminths and blocking glucose uptake. This action leads to eventual death of the worm, after which it can be passed through the feces.
- Ivermectin: Ivermectin is an interesting antiparasitic agent because it has activity as both a pediculicide and an antihelminth. When used topically, it can treat lice and scabies infestations; when taken orally, it can treat a variety of helminth infections. Ivermectin works by inhibiting parasite nerve and muscle tissue, causing paralysis and death of the organism.
Table 7.15 lists antiparasitic and anthelminthic drugs and typical routes and dosing for adult clients.
| Drug | Routes and Dosage Ranges |
|---|---|
| Lindane | Head lice: Apply 30–60 mL shampoo to dry hair and massage into hair for 4 min; add small quantities of water to hair until lather forms, then rinse hair and comb with fine-tooth comb to remove nits. |
|
Mebendazole
( Emverm ) |
Ascariasis: 100 mg orally twice daily for 3 days or 500 mg orally once daily. |
|
Metronidazole
( Flagyl ) |
Giardiasis:
250 mg orally 3 times daily for 5–7 days.
Trichomoniasis: 500 mg orally twice daily for 7 days. Amebiasis: 500 mg orally every 8 hours for 7–10 days. |
|
Nitazoxanide
( Alinia ) |
Giardiasis: 500 mg orally every 12 hours for 3 days. |
|
Permethrin
( Elimite ) |
Head lice: Apply enough lotion or cream to saturate hair. Leave on for 10 min and then rinse with warm water. |
|
Spinosad
( Natroba ) |
Head lice: Apply sufficient amount to cover scalp. |
Adverse Effects and Contraindications
Most oral antiprotozoal and anthelminthic drugs can cause gastrointestinal upset. Any topical scabicides and pediculicides may cause skin irritation to the area where they are applied. Excessive use of lindane is not recommended in high doses, especially in children, because it can cause central nervous system excitation and seizure. Malathion should be kept away from children because ingesting it could cause cholinergic poisoning.
Table 7.16 is a drug prototype table for antiparasitic and anthelminthic drugs featuring metronidazole. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Nitroimidazole Mechanism of Action Causes loss of helical DNA structure and strand breakage to cause cell death |
Drug Dosage
Giardiasis: 250 mg orally 3 times daily for 5–7 days. Trichomoniasis: 500 mg orally twice daily for 7 days. Amebiasis: 500 mg orally every 8 hours for 7–10 days. |
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Indications
Amebicide Antiprotozoal Therapeutic Effects Reduces symptoms of infection |
Drug Interactions
Disulfiram Fosphenytoin Haloperidol Mebendazole Food Interactions Ethanol |
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Adverse Effects
Nausea Vaginitis Headache Abdominal pain Diarrhea Xerostomia Dizziness |
Contraindications
Hypersensitivity Caution: Superinfection Hepatic impairment Renal impairment Seizure disorder |
Nursing Implications
The nurse should do the following for clients who are taking antiparasitic or anthelminthic drugs:
- Monitor for signs and symptoms of anaphylaxis (e.g., shortness of breath, difficulty breathing, difficulty swallowing).
- Advise the client to take the entire prescribed course of the medication to ensure adequate treatment and to reduce the development of drug resistance.
- Instruct the client to maintain adequate hydration; monitor kidney function for renally eliminated medications.
- Provide client teaching regarding the drug and when to call the health care provider. See below for client teaching guidelines.
Client Teaching Guidelines
The client taking an antiparasitic or anthelminthic drug should:
- Alert their health care provider about any signs of allergic reactions including throat swelling, severe itching, rash, or chest tightness.
- Alert their health care provider that they are taking these medications, including the dose and frequency.
- Take the drug with food if it causes an upset stomach.
- Take a missed dose as soon as they remember; however, they should not take double doses.
- Avoid taking tinidazole or metronidazole with alcohol because the interaction can cause a severe flushing reaction.
FDA Black Box Warning
Lindane Lotion
Seizures and deaths have been reported following repeated or prolonged use and in rare cases following a single application.


8.8: Applications to Early Education
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Understanding how children think and learn has proven useful for improving education. Activities like playing games that involve working with numbers and spatial relationships can give young children a developmental advantage over peers who have less exposure to the same concepts.
Even before they enter kindergarten, the mathematical knowledge of children from low-income backgrounds lags far behind that of children from more affluent backgrounds. Ramani and Siegler (2008) hypothesized that this difference is due to the children in middle- and upper-income families engaging more frequently in numerical activities, for example playing numerical board games such as Chutes and Ladders. Chutes and Ladders is a game with a number in each square; children start at the number one and spin a spinner or throw a dice to determine how far to move their token. Playing this game seemed likely to teach children about numbers, because in it, larger numbers are associated with greater values on a variety of dimensions. In particular, the higher the number that a child’s token reaches, the greater the distance the token will have traveled from the starting point, the greater the number of physical movements the child will have made in moving the token from one square to another, the greater the number of number-words the child will have said and heard, and the more time will have passed since the beginning of the game. These spatial, kinesthetic, verbal, and time- based cues provide a broad-based, multisensory foundation for knowledge of numerical magnitudes (the sizes of numbers), a type of knowledge that is closely related to mathematics achievement test scores (Booth & Siegler, 2006).
Playing this numerical board game for roughly 1 hour, distributed over a 2-week period, improved low-income children’s knowledge of numerical magnitudes, ability to read printed numbers, and skill at learning novel arithmetic problems. The gains lasted for months after the game-playing experience (Ramani & Siegler, 2008; Siegler & Ramani, 2009). An advantage of this type of educational intervention is that it has minimal if any cost—a parent could just draw a game on a piece of paper.
Cognitive developmental research has shown that phonemic awareness—that is, awareness of the component sounds within words—is a crucial skill in learning to read. To measure awareness of the component sounds within words, researchers ask children to decide whether two words rhyme, to decide whether the words start with the same sound, to identify the component sounds within words, and to indicate what would be left if a given sound were removed from a word. Kindergartners’ performance on these tasks is the strongest predictor of reading achievement in third and fourth grade, even stronger than IQ or social class background (Nation, 2008). Moreover, teaching these skills to randomly chosen 4- and 5-year-olds results in their being better readers years later (National Reading Panel, 2000).
Understanding of cognitive development is advancing on many different fronts. One exciting area is linking changes in brain activity to changes in children’s thinking (Nelson et al., 2006). Although many people believe that brain maturation is something that occurs before birth, the brain actually continues to change in large ways for many years thereafter. For example, a part of the brain called the prefrontal cortex, which is located at the front of the brain and is particularly involved with planning and flexible problem solving, continues to develop throughout adolescence (Blakemore & Choudhury, 2006). Such new research domains, as well as enduring issues such as nature and nurture, continuity and discontinuity, and how to apply cognitive development research to education, insure that cognitive development will continue to be an exciting area of research in the coming years. 29