Chapter 14.6, Problem 24ES

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

If G is the solid in Exercise 23 and $f\left(x,y,z\right)$ is continuous on G, then ${\displaystyle \underset{G}{\iiint }f\left(x,y,z\right)dV={\displaystyle {\int }_0^{\pi /4}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_1^{3}F\left(\rho ,\theta ,\varphi \right)}}}}{\rho }^2\sin \varphi d\rho d\theta d\varphi$ where $F\left(\rho ,\theta ,\varphi \right)=f\left(\rho \sin \varphi \sin \theta ,\rho \sin \varphi \cos \theta ,\rho \cos \varphi \right).$