2 min readβ’june 18, 2024

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:Β

Letβs 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.Β

Hereβs a basic formula to understand the concept:Β

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.Β Β

If youβre still confused, try out this example and see how you do.Β

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:

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