# Finding the Area of a Surface of Revolution

The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.

To find the area of a surface of revolution between *a* and *b**,* use the following formula:

This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it. The integral is made from two pieces:

The arc-length formula, which measures the length along the surface

The formula for the circumference of a circle, which measures the length around the surface

So multiplying these two pieces together is similar to multiplying length and width to find the area of a rectangle. In effect, the formula allows you to measure surface area as an infinite number of little rectangles.

When you’re measuring the surface of revolution of a function *f*(*x*) around the *x*-axis, substitute *r* = *f*(*x*) into the formula:

For example, suppose that you want to find the area of revolution that’s shown in this figure.

*y*=

*x*

^{3}between

*x*= 0 and

*x*= 1.

To solve this problem, first note that for

So set up the problem as follows:

To start off, simplify the problem a bit:

You can solve this problem by using the following variable substitution:

Now substitute *u* for 1+ 9*x*^{4} and

for *x*^{3} *dx *into the equation:

Notice that you change the limits of integration: When *x* = 0, *u* = 1. And when *x* = 1, *u* = 10.

Now you can perform the integration:

Finally, evaluate the definite integral: