Surfaces of revolution are one of many applications of integration. A surface of revolution is generated when a region bounded by two or more functions is rotated about an axis. Generally, this will be either the x or y axis, but it can be any axis. The surface area of a surface of revolution is the most useful thing to find, and it can be found easily using a few simple procedures.

  1. Graph the function. This will help you determine the points of intersection, as well as which graph is the top-most or right-most.
  2. Determine where the two functions that bind the region intersect.
  3. Determine the radius function, r(x). r(x) = f(x) - g(x), where f(x) is either the top-most (revolution about the x axis or an axis parallel to the x axis) or right-most (revolution about the y axis or an axis parallel to the y axis), and g(x) is either the bottom-most or left-most.
  4. Determine the interval on which you will integrate. Usually, this information will be given in terms of x, but if you must integrate with respect to y, make sure to change your terms by passing the interval through f(x)!
  5. Find the surface area using the formula (for surfaces generated about a verticle axis) S = 2pi*Sab (r(x) + sqrt(1 + (dy/dx)^2))dx. Should you be integrating with respect to y (i.e. your surface of revolution has been rotated about a horizontal axis), use 2pi*Sab (r(y) + sqrt(1 + (dx/dy)^2)dy. Keep in mind that sqrt(1 + (dy/dx)^2) and sqrt(1 + (dx/dy)^2) are, when integrated on the closed interval [a,b] and [c,d], are arc lengths.