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:

**CITE THIS AS:**

**Stephanie Glen**. "Surface of Revolution" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/surface-of-revolution/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!