8.7 Volumes with Cross Sections: Squares and Rectangles

Now that you can find the area between curves, you should have no problem finding the volumes of 3-dimensional shapes. In order to find the area of a shape using a square cross section, take a look at the example below. 

Let’s say we have a function x = 4 - y^2 and we know that the cross sections are squares. Because the height and width of squares are the same, finding the volume is quite easy. Before, we were finding the area by adding up a bunch of infinitely thin slices. Now, we are finding the volume by adding up a bunch of infinitely thin squares.

The area of one square cross section is ((4 - y^2) - -(4 - y^2))^2. This can be simplified to (2(4 - y^2))^2. This is a simple right minus left represented in topic 8.5. Now we integrate from the intersection points found by setting y = 0.

Like before with squares, we can use rectangles to find the volume of an object. The area of a rectangle is A = lw, where l is length and w is width. In each case, once we figure out l and w, we can integrate between our bounds to find the volume.

We're given the function y = x^3/2 and told to find a volume with rectangular cross sections perpendicular to the y-axis. This is an important distinction. This means that the length of each rectangle is NOT x^3/2 (y), but rather (2y)^(1/3) [this is obtained by solving for x], since our rectangles are in terms of y, not x. Now, we just need to find l. Luckily, we're simply given this in the problem. Therefore, our integral will be the integral from 0 to 4 of (2y)^(1/3) * y/2.