Area of Surface of Revolution Example
Situation: You are required to find the area of the surface formed by revolving the graph of f(x) = x3 on the interval [0, 1] about the x-axis, seen below.
Example

Solution: The distance between the x-axis and the graph of f is r(x) = f(x), and becuase f '(x) = 3x2, the surface area is
               b
S = 2pi §  r(x)SQRT(1 + [f '(x)]2) dx
               a
            1
= 2pi §  x3SQRT(1 + (3x2)2) dx
            0
                   1
= (2pi)/36 §  (36x3)(1 + 9x4)1/2 dx
                   0
                                            1
= (pi/18)[(1 + 9x4)3/2/(3/2)]
                                            0
= pi/27(103/2 - 1)

= 3.563