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, watch this video tutorial or follow the steps below:
![image0.png](https://www.dummies.com/wp-content/uploads/314778.image0.png)
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:
![image1.png](https://www.dummies.com/wp-content/uploads/314779.image1.png)
For example, suppose that you want to find the area of revolution that’s shown in this figure.
![Measuring the surface of revolution of <i>y</i> = <i>x</i><sup>3</sup> between <i>x</i> = 0 and <i>](https://www.dummies.com/wp-content/uploads/314780.image2.jpg)
To solve this problem, first note that for
![image3.png](https://www.dummies.com/wp-content/uploads/314781.image3.png)
So set up the problem as follows:
![image4.png](https://www.dummies.com/wp-content/uploads/314782.image4.png)
To start off, simplify the problem a bit:
![image5.png](https://www.dummies.com/wp-content/uploads/314783.image5.png)
You can solve this problem by using the following variable substitution:
![image6.png](https://www.dummies.com/wp-content/uploads/314784.image6.png)
Now substitute u for 1+ 9x4 and
![image7.png](https://www.dummies.com/wp-content/uploads/314785.image7.png)
for x3 dx into the equation:
![image8.png](https://www.dummies.com/wp-content/uploads/314786.image8.png)
Notice that you change the limits of integration: When x = 0, u = 1. And when x = 1, u = 10.
![image9.png](https://www.dummies.com/wp-content/uploads/314787.image9.png)
Now you can perform the integration:
![image10.png](https://www.dummies.com/wp-content/uploads/314788.image10.png)
Finally, evaluate the definite integral:
![image11.png](https://www.dummies.com/wp-content/uploads/314789.image11.png)