Chapter 14.6, Problem 22ES

Determine whether the statement is true or false. Explain your answer.

If $0\le {\rho }_1<{\rho }_2,0\le {\theta }_1<{\theta }_2\le 2\pi ,\text{and}0\le {\varphi }_1<{\varphi }_2\le \pi ,$ then the volume of the spherical wedge bounded by the spheres $\rho ={\rho }_1\text{and}\rho ={\rho }_2,$ the half-planes $\theta ={\theta }_1\text{and}\theta ={\theta }_2,$ and the cones $\varphi ={\varphi }_1\text{and}\varphi ={\varphi }_2$ is ${\displaystyle {\int }_{{\theta }_1}^{{\theta }_2}{\displaystyle {\int }_{{\varphi }_1}^{{\varphi }_2}{\displaystyle {\int }_{{\rho }_1}^{{\rho }_2}{\rho }^2\sin \varphi d\rho d\varphi d\theta }}}$