Surface of Revolution

Calculus Definitions >

A surface of revolution is the surface that you get when you rotate a two dimensional curve around a specific axis.

The image below shows a function f(x) over an interval [a,b], and the surface of revolution you get when you rotate it around the x axis.

• If your line isn’t parallel to the axis it is rotated around, the surface of revolution will be a cone, or a part of a cone.
• If your curve is a straight line, parallel to the axis you are rotating around, your surface of revolution will be a cylinder.
• The surface of revolution of a line perpendicular to the axis will just be a circle.

Finding the Area of a Surface of Revolution.

You can use calculus to find the area of a surface of revolution. Suppose the curve is described by two parametric functions x(t) and y (t); you want to find the surface that results when the segment of that curve ranging from x = a to x = b is rotated around the y axis. Then, so long as x(t) is not negative on the interval, the area of the surface you generate will be:

This general formula can be specialized to situations you are likely to meet in math problem sets now or in engineering applications later. For example, if you have a curve represented by y = f(x), a ≤ x ≤ b, and you want to revolve around the x axis, you can simplify your surface area integral as:


If you’re revolving around the y axis, on the interval a ≤ y ≤ b, the integral becomes:

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