Chapter 5.5, Problem 274E

a. A function F is said to have spherical svmmetiy if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as F(x. y. z) = f(p). where $\rho =\sqrt{x^2+y^2+z^2}$ .

Show that ${\displaystyle \underset{B}{\iiint }F\left(x,y,z\right)dV=2\pi {\displaystyle \underset{a}{\overset{b}{\int }}{\rho }^{2}f\left(\rho \right)d\rho ,}}$where B is the region between the upper concentric hemispheres of radii a and b centered at the origin, with 0 < a < b and F a spherical function defined on B.

b. Use the previous result to show that ${\displaystyle \underset{B}{\iiint }\left(x^2+y^2+z^2\right)\sqrt{x^2+y^2+z^2}dV}=21\pi$ where $B=\left\{\left(x,y,z\right)|1\le x^2+y^2+z^2\le 2,z\ge 0\right\}$