Chapter 5.5, Problem 247E

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. , z) = f(r) + Iz(z). where f and Ii are b differentiable functions. If ff(r)dr = 0 and Iz(0) = 0. where f and Ii are antiderivatives of f and Ii. respectively, show that I F(x. y, z)dV = 2xc(bf(b) — af(a)) + x(b2 — a2)h(c).

b. Use the previous result to show that Ill (z + sinR ‘(hdvd = 6,r2(,r —2), where B is a cylindrical shell with inner radius n. outer radius 2jr, and height 2.