Chapter 15.8, Problem 49E

(a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere $r^2+z^2=a^2$ and below by the cone $z=r\cot {\varphi }_0\left(\text{or}\varphi ={\varphi }_0\right)$, where $0<{\varphi }_0<\pi /2$, is

$V=\frac{2\pi a^3}{3}\left(1-\cos {\varphi }_0\right)$

(b) Deduce that the volume of the spherical wedge given by ${\rho }_1\le \rho \le {\rho }_2,{\theta }_1\le \theta \le {\theta }_2\text{, }{\varphi }_1\le \varphi \le {\varphi }_2,$ is

$\Delta V=\frac{\rho \stackrel{3}{2}-\rho \stackrel{3}{1}}{3}\left(\cos {\varphi }_1-\cos {\varphi }_2\right)\left({\theta }_2-{\theta }_1\right)$

(c) Use the Mean Value Theorem to show that the volume in part (b) can be written as

$\Delta V=\stackrel{~2}{\rho }\sin \tilde{\varphi }\Delta \rho \Delta \theta \Delta \varphi$

Where $\tilde{\rho }$ lies between ${\rho }_1$ and ${\rho }_2$, $\tilde{\varphi }$ lies between ${\varphi }_1$ and ${\varphi }_2\Delta \rho ={\rho }_2-{\rho }_{1,}\Delta \theta ={\theta }_2-{\theta }_1,\text{ and }\Delta \varphi \text{=}{\varphi }_2-{\varphi }_1$