Let B be a cylindrical shell with inner radius a, outer radius b, and height c, where 00. Assume that a function F defined on B can be expressed in cylindrical coordinates as F(x, y, z)=f(r) g(theta) h(z), where f, g, and h are differentiable functions. If int_aᵇ tildef(r) dr=0, where tildef is an antiderivative of f, show that iiint F(x, y, z) d V=[b tildef(b)-a tildef(a)][ tildeg(2 pi)- tildeg(0)][ tildeh(c)- tildeh(0)], where tildeg and tildeg are antiderivatives of g and h, respectively.
