<s>[INST] Find the volume of the part of the cylinder $\frac{x^2}{a^2}+\frac{z^2}{c^2}=1$ which lies between the plane $y=0 ,y=mx$ Find the volume of the part of the cylinder $\frac{x^2}{a^2}+\frac{z^2}{c^2}=1$ which lies between the plane $y=0, y=mx$
The limits of integration which I set went as follows:
$${\int \int}_{D} \int_{y=0}^{mx}\mathrm dy\mathrm dx\mathrm dz$$ where $$D=\big \lbrace \: (x,z) \mid \frac{x^2}{a^2}+\frac{z^2}{c^2}=1 \: \big \rbrace$$
Now let $x=au,z=bv$, then $D$ transforms into $\big \lbrace \: (u,v) \mid u^2 + v^2=1 \: \big \rbrace$
Then the integral becomes $$\int_{u=-1}^1 \int_{v=-\sqrt{1-u^2}}^{v=\sqrt{1-u^2}} ma^2cu\mathrm du\mathrm dv$$
 [/INST] From $ \dfrac{x^2}{a^2} + \dfrac{z^2}{c^2} = 1 $, take
$x = a u \cos t$
$z = c u \sin t$
$y = y$
where $ 0 \le t \le 2 \pi $ and $ 0 \le u \le 1 $
The Jacobian of the above transformation is
$ J = \dfrac{\partial(x, y, z) }{\partial(u, t, y)} = \big| \begin{vmatrix} a \cos t && - a u \sin t && 0 \\ 0 && 0 && 1 \\ c \sin t && c u \cos t && 0 \end{vmatrix} \big| = a c u$
Now the volume is given by the following integral
$\begin{equation} \begin{split}
 V &= \displaystyle \int_{0}^{2\pi} \int_0^1 \int_{y=0}^{a m u |\cos t|} (a c u) \ dy \ du \ dt =\int_{0}^{2\pi} \int_0^1 (a^2 m c u^2) |\cos t | \ du \ dt \\  &= \dfrac{1}{3} a^2 c m \int_{0}^{2 \pi} | \cos t | dt = \dfrac{4}{3} a^2 c m \end{split} \end{equation}$
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