Chapter 5.5, Problem 275E

a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates ( $\rho ,\theta ,\phi$ ) is $F\left(x,y,z\right)=f\left(\rho \right)\cos \phi$ . Show that if $g\left(a\right)=g\left(b\right)=0$ and ${\displaystyle \underset{a}{\overset{b}{\int }}h\left(h\rho \right)d\rho =0}$ . then ${\displaystyle \underset{B}{\iiint }F\left(x,y,z\right)dV=\frac{{\pi }^2}{4}}\left[ah\left(a\right)-bh\left(b\right)\right]$where g is an antiderivative of f and h is an antiderivative of g.

b. Use the previous result to show that ${\displaystyle \underset{B}{\iiint }\frac{z\cos \sqrt{x^2+y^2+z^2}}{\sqrt{x^2+y^2+z^2}}}dV=\frac{3\pi }{2}$. where B is the region between the upper concentric hemispheres of radii $\pi$ and 2 $\pi$ centered at the origin and situated in the first octant.