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 arclength 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 xaxis, 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.