To find an area between two functions, you need to set up an equation with a combination of definite integrals of both functions. For example, suppose that you want to calculate the shaded area between y = x2 and
![image0.png](https://www.dummies.com/wp-content/uploads/312063.image0.png)
as shown in this figure.
![image1.jpg](https://www.dummies.com/wp-content/uploads/312064.image1.jpg)
First, notice that the two functions y = x2 and
![image2.png](https://www.dummies.com/wp-content/uploads/312065.image2.png)
intersect where x = 1. This information is important because it enables you to set up two definite integrals to help you find region A:
![image3.png](https://www.dummies.com/wp-content/uploads/312066.image3.png)
Although neither equation gives you the exact information that you’re looking for, together they help you out. Just subtract the second equation from the first as follows:
![image4.png](https://www.dummies.com/wp-content/uploads/312067.image4.png)
With the problem set up properly, now all you have to do is evaluate the two integrals:
![image5.png](https://www.dummies.com/wp-content/uploads/312068.image5.png)
So the area between the two curves is
![image6.png](https://www.dummies.com/wp-content/uploads/312069.image6.png)
As another example, suppose that you want to find the area between y = x and
![image7.png](https://www.dummies.com/wp-content/uploads/312070.image7.png)
as shown in this figure.
![image8.jpg](https://www.dummies.com/wp-content/uploads/312071.image8.jpg)
This time, the shaded area is two separate regions, labeled A and B. Region A is bounded above by
![image9.png](https://www.dummies.com/wp-content/uploads/312072.image9.png)
and bounded below by y = x. However, for region B, the situation is reversed, and the region is bounded above by y = x and bounded below by
![image10.png](https://www.dummies.com/wp-content/uploads/312073.image10.png)
Regions C and D are also labeled, as they both figure into the problem.
The first important step is finding where the two functions intersect — that is, where the following equation is true:
![image11.png](https://www.dummies.com/wp-content/uploads/312074.image11.png)
Fortunately, it’s easy to see that x = 1 satisfies this equation.
Now you want to build a few definite integrals to help you find the areas of region A and region B. Here are two that can help with region A:
![image12.png](https://www.dummies.com/wp-content/uploads/312075.image12.png)
Notice that the second definite integral is evaluated without calculus, using simple geometry. This is perfectly valid and a great timesaver.
Subtracting the second equation from the first provides an equation for the area of region A:
![image13.png](https://www.dummies.com/wp-content/uploads/312076.image13.png)
Now build two definite integrals to help you find the area of region B:
![image14.png](https://www.dummies.com/wp-content/uploads/312077.image14.png)
This time, the first definite integral is evaluated by using geometry instead of calculus. Subtracting the second equation from the first gives an equation for the area of region B:
![image15.png](https://www.dummies.com/wp-content/uploads/312078.image15.png)
Now you can set up an equation to solve the problem:
![image16.png](https://www.dummies.com/wp-content/uploads/312079.image16.png)
At this point, you’re forced to do some calculus:
![image17.png](https://www.dummies.com/wp-content/uploads/312080.image17.png)
The rest is just arithmetic:
![image18.png](https://www.dummies.com/wp-content/uploads/312081.image18.png)