Chapter 8.3, Problem 48E

The Second Theorem of Pappus is in the same spirit as Pappus’s Theorem on page 565. but for surface area rather than volume: Let C be a curve that lies entirely on one side of a line l in the plane. If C is rotated about l, then the area of the resulting surface is the product of the arc length of C and the distance traveled by the centroid of C (see Exercise 47).

(a) Prove the Second Theorem of Pappus for the case where C is given by y = f(x), f(x) ≥ 0. and C is rotated about the x-axis.

(b) Use the Second Theorem of Pappus to compute the surface area of the half-sphere obtained by rotating the curve from Exercise 47 about the x-axis. Does your answer agree with the one given by geometric formulas?