Chapter 14.6, Problem 30ES

In this exercise we will obtain a formula for the volume of the spherical wedge illustrated in Figures 14.6.7 and 14.6.9.

(a) Use a triple integral in cylindrical coordinates to show that the volume of the solid bounded above by a sphere $\rho ={\rho }_0,$ below by a cone $\varphi ={\varphi }_0,$ and on the sides by $\theta ={\theta }_1\text{and}\theta ={\theta }_2\left({\theta }_1<{\theta }_2\right)$ is $V=\frac{1}{3}{\rho }_0^3\left(1-\cos {\varphi }_0\right)\left({\theta }_2-{\theta }_1\right)$

(b) Subtract appropriate volumes and use the result in part (a) to deduce that the volume $\Delta V$ of the spherical wedge is $\Delta V=\frac{{\rho }_2^3-{\rho }_1^3}{3}\left(\cos {\varphi }_1-\cos {\varphi }_2\right)\left({\theta }_2-{\theta }_1\right)$

(c) Apply the Mean-Value Theorem to the functions $\cos \varphi \text{and}{\rho }^3$ to deduce that the formula in part (b) can be written as $\Delta V={\rho }^{*2}\sin \varphi *\Delta \rho \Delta \varphi \Delta \theta$ where $\Delta \rho ={\rho }_2-{\rho }_1,\Delta \varphi ={\varphi }_2-{\varphi }_1,\Delta \theta ={\theta }_2-{\theta }_1,$ and ${\rho }^{*}$ are between ${\rho }_1\text{and}{\rho }_2,{\varphi }^{*}$ is between ${\varphi }_1\text{and}{\varphi }_2.$