Chapter 5.5, Problem 248E

a. Let B be a cylindrical shell with inner radius a. outer radius b. and height c, where 0 < a < b and c> 0. Assume that a function F defined on B can be expressed in cylindrical coordinates as F(x. v z) = f(r)g( $\theta$)h(z). where f. g. and Ii are differentiable functions. If ${\displaystyle \underset{a}{\overset{b}{\int }}f}-\left(r\right)dr=0$ . where is an antiderivative of f. show that J1 F(x. y. z)dV = [b?(b) — af(a)]I(2x) — — 11(0)]. where and Ii are antiderivatives of g and Ii. respectively.

b. Use the previous result to show that z sinLt + v2dx dv dz = —1 2,r2. where B B is a cylindrical shell ‘with inner radius n’, outer radius 2,r. and height 2.