## Calculus 8th Edition

$V=\frac{7\pi}{6}$ One-eighth of a hollow sphere with outer radius $2$ and inner radius $1$.
The $\rho^{2}sin\phi d\rho d\theta d \phi$ tells us that we are dealing with spherical co-ordinates. The radius ranges from $1$ to $2$ , while the angle bounds limits us to one octant , or one-eighth of the hollow sphere. $=\int_{0}^{\pi/2}\int_{0}^{\pi/2}(\frac{\rho^{2}}{3}sin\phi)|_{1}^{2}d\phi d\theta$ $=\int_{0}^{\pi/2}\int_{0}^{\pi/2}\frac{7}{3}sin(\phi)d\phi d\theta$ $=\int_{0}^{\pi/2}(-\frac{7}{3})cos(\phi)|_{0}^{\pi/2}$ $=\int_{0}^{\pi/2}(\frac{7}{3})d \theta$ $=\frac{7\pi}{6}$ Hence, $V=\frac{7\pi}{6}$ One-eighth of a hollow sphere with outer radius $2$ and inner radius $1$.