<s>[INST] Area Between Two Curves I have been trying to solve a homework problem that reads: "The region between the graphs of $f(x) = x^{2}+2$ and $g(x) = -5x+2$ has what area?" I have been trying to solve this problem and have come up with the answer of $185/6$ square units, but it has come to my attention that that is not correct. So, I was hoping maybe someone can help me out with this problem. Thank you so much! 
 [/INST] The two graphs intersect at the points where their $x$-coordinates are equal and their $y$-coordinates are equal.  Their $y$-coordinates are $x^2+2$ and $-5x+2$.  These are equal when
$$
x^2+2=-5x+2.
$$
Hence $x^2=-5x$.  If $x\ne 0$ then we can divide both sides by $x$ and get $x=-5$.  If $x=0$, then that is also a solution, as seen by substitution: $0^2 \overset{\text{?}}{=} -5\cdot 0$.
When $x$ is between $-5$ and $0$, then $x^2+2$ is smaller than $-5x+2$, as seen by plugging in any number between $-5$ and $0$.  So
$$
\begin{align}
& \phantom{{}=} \int_{-5}^0 (\text{bigger function minus smaller function}) \\[8pt]
& = \int_{-5}^0 (-5x+2)-(x^2+2) \,dx = \int_{-5}^0 -5x -x^2 \,dx \\[8pt]
& = \left.\frac{-5x^2}{2} -\frac{x^3}{3}\right|_{x=-5}^{x=0} = \frac{125}{2} -\frac{125}{3} = \frac{125}{6}.
\end{align}
$$
</s>