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.

Measuring the surface of revolution of y = x3 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+ 9x4 and

for x3 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: