## Calculus (3rd Edition)

$$\frac{4802\pi }{5}$$
Given $$x=4-y,\ \ x=16-y^2$$ We first find the intersection points \begin{align*} 4-y&=16-y^2\\ y^2-y -12&=0 \end{align*} Then $y=-3,\ 4$ Then the volume of the solid obtained by rotating the region enclosed by the graph of $x=4-y,\ \ x=16-y^2$ about the $y-$axis is given by \begin{align*} V&=\pi\int_{-3}^{4}[f(y)]^2-[g(y)]^2dy\\ &=\pi \int_{-3}^{4}\left[\left(16-y^{2}\right)^{2}-(4-y)^{2}\right] d y\\ &=\pi \int_{-3}^{4}\left[y^{4}-33 y^{2}+8 y+240\right] d y\\ &=\pi \left(\frac{y^{5}}{5}-\frac{33 y^{3}}{3}+\frac{8 y^{2}}{2}+240 y\right)\bigg|_{-3}^{4}\\ &=\frac{4802\pi }{5} \end{align*} We use the Mathematica program to plot the solid.