Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below.

Question

Quizplus · Accepted Answer

The solid is bounded by the graphs of $z=4-2x$ and $z=x^{2}+y^{2}$. Since we want to find volume, we will integrate $dz$ first. The limits of $z$ are from $4-2x$ to $x^{2}+y^{2}$.
$$\int_{0}^{2\pi}\int_{0}^{1}\int_{4-2r\cos{	heta}}^{r^{2}}r dz\,dr\,d	heta = \int_{0}^{2\pi}\int_{0}^{1}\frac{r^3}{3}-r(2\cos	heta)\,dr\,d	heta=\frac{\pi}{2}$$
Therefore, the answer is choice C.

Set Up and Evaluate a Double Integral to Find the Volume