8.5 Finding the Area Between Curves Expressed as Functions of y

This is similar to Topic 8.4 except we are now using curves expressed as functions of y instead of x. If the same function is both the top and the bottom of the slice (like the example below), we must use horizontal slices instead of vertical ones. 

Let’s say we want to find the area between the curve x = y^2 and the curve x = y + 6 from y = 0 to y = 3. When you slice up the section horizontally, the right edge of the area you’re trying to find is the curve x = y + 6 and the left edge is the curve x = y^2. Thus, you would integrate (y + 6 - y^2) from 0 to 3 to find the area. 

It is important to note that your functions must be expressed in terms of y to use horizontal slicing. Also, your endpoints must be expressed as functions of y. In the example above, the endpoints are y = 0 and y = 3.