This is one of the most important topics of Unit 8. In this topic, we will discuss how to find the area between two curves expressed as functions of x. This topic will set you up to understand more complex topics moving forward. To understand how to find the area, take a look at this simple example:
Let’s say we want to find the area between the curve y = x and y = x^2 from x = 2 to x = 4. In order to find the area, you can imagine we are slicing the region vertically, into a bunch of infinitely thin slices. The area would be the sum of all the slices. To add all the slices, you can use a definite integral. Integrate the function (x^2 - x) from 2 to 4.
Here’s a basic formula to understand the concept:
Also, it is important to mention that you can only use this specific method when your functions are expressed in terms of x. In the example above, our functions were x and x^2. If the functions were y and y^2, we would have to use a slightly different approach. To learn more, look at Topic 8.5: Finding the Area Between Curves Expressed as Functions of y.
If you’re still confused, try out this example and see how you do.
Question: Find the area between the functions y = x^2 - 2 and -x^2.
Solution: First, we should set the functions equal to each other to find the intersection points. If you do x^2 - 2 = -x^2 and solve for x, you should get x = -1 and x = 1. Next, graph the functions to figure out which one is on top. You should get something like this: