Chapter 6.6, Problem 42E

Surface plus cylinder Suppose f is a nonnegative function with a continuous first derivative on [a, b]. Let L equal the length of the graph of f on [a, b] and let S be the area of the surface generated by revolving the graph of f on [a, b] about the x-axis. For a positive constant C, assume the curve y = f(x) + C is revolved about the x-axis. Show that the area of the resulting surface equals the sum of S and the surface area of a right circular cylinder of radius C and height L.