Volume Using Known Cross Sections - Calculus 2

Previously we have seen how to use the the disk method (among other methods) to calculate the volume of solids of revolution with definite integrals. However, we are not limited to calculating the volume of solids of revolution, or solids with circular cross sections. We can also use definite integrals to calculate the volume of other types of solids. Specifically, we can calculate the volume of solids with known cross sections, or other familiar shapes besides circles, that have area that is simple to determine.

These common cross sections include squares, rectangles, semicircles, and more! When a solid with a base formed by a function has cross sections such as these, we can integrate the cross sectional area to find the volume of the solid. For the disk method, the cross sectional area was the area of a circle. But  now the cross sectional area will be the area of other shapes, and so we will need to know the area formulas for those shapes.

Additionally, whether the known cross sections are perpendicular to the x-axis or y-axis is important in determining how to set up a definite integral that represents the volume. It will be the deciding factor in whether we integrate with respect to x, or with respect to y, as well as how to express the cross sectional area in terms of one of these variables.

In this lesson, you will learn:

•  The common cross sections of solids and their area formulas used for calculating the volume of the solids using definite integrals.
• How to calculate the volume of solids with known cross sections perpendicular to the x-axis using definite integrals
• How to calculate the volume of solids with known cross sections perpendicular to the y-axis using definite integrals

I hope you find this video to be helpful!

-Josh