Chapter 17.6, Problem 72E

Surfaces of revolution Suppose y = f(x) is a continuous and positive function on [a, b]. Let S be the surface generated when the graph of f on [a, b] is revolved about the x-axis.

a.    Show that S is described parametrically by r(u, v) = 〈u, f(u) cos v, f(u) sin v〉, for a ≤ u ≤ b, 0 ≤ v ≤ 2 π.

b.    Find an integral that gives the surface area of S.

c.    Apply the result of part (b) to find the area of the surface generated with f(x) = x³, for 1 ≤ x ≤ 2.

d.    Apply the result of part (b) to find the area of the surface generated with f(x) = (25 – x²)^1/2, for 3 ≤ x ≤ 4.