Chapter 6, Problem 16RE

Let C be the curve $27x-y^3=0$ between $y=0$ and $y=2$ . In each part, set up, but do not evaluate, an integral or a sum of integrals that solves the problem.

(a) Find the area of the surface generated by revolving $C$ about the $x\text{-axis}$ by integrating with respect to $x$ .

(b) Find the area of the surface generated by revolving $C$ about the $y\text{-axis}$ by integrating with respect to $y$ .

(c) Find the area of the surface generated by revolving $C$ about the fine $y=-2$ by integrating with respect to $y$ .