Answer to Question #289096 in Calculus for jea

Question

Find the volume of the solid S enclosed by the cylinder x2 + y2 = 9 and the planes y + z = 5 and

 = 1. (Hint: Convert to Cylindrical/Polar Coordinates)

Answer 1

$$V=\int \int \int dV=\displaystyle\int_{-3}^{3}\displaystyle\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\displaystyle\int_{1}^{5-y}dzdydx$$

$$=\displaystyle\int_{-3}^{3}\displaystyle\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}(4-y)dydx$$

$$=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{3}(4-r\sin \theta)rdrd\theta$$

$$=\displaystyle\int_{0}^{2\pi}[2r^2-\dfrac{r^3}{3}\sin \theta]\begin{matrix} 3 \\ 0 \end{matrix}d\theta$$

$$=\displaystyle\int_{0}^{2\pi}(18-9\sin \theta)d\theta$$

$$=[18\theta+9\cos \theta]\begin{matrix} 2\pi \\ 0 \end{matrix}=36\pi ({units}^3)$$

$36\pi$ cubic units.