Finding the Volumes of Solids With Known Cross Sections

Instructor: Cameron Smith

Cameron has a Master's Degree in education and has taught HS Math for over 25 years.

In this lesson, you will learn how a solid with a known cross section is created. Then, you will learn how to set up the integral that helps you find the volume of the solid with the known cross section.

Volumes of Solids with Known Cross-Sections

You can use integrals to find volumes of different kinds of objects. In this lesson, you will learn how to find the volume of a solid object that has known cross sections, or in other words cross sections that are known figures. Those cross sections could be squares, rectangles, semicircles, or any other geometric figure. You will learn how to set up the integral used to find the volumes of these kinds of solids, and then you can evaluate the integral to find the volume of the solid.

It is helpful to remember that an integral is a way of adding up infinite sums. When you find the area under a curve, you are finding the sum of the area of infinite rectangles. You are going to use a similar approach in this lesson by setting up an integral that finds the sum of infinite volumes, that equal the volume of the solid.

The solids you are going to create, have a base that is a bounded region in the xy-plane. This base could be a circle centered at the origin, or the region between graphs. The solid is built with cross sections that are perpendicular to that base and either the x or y-axis. In the picture the cross sections are squares built on the area between the graph f(x) = x^2 and the x-axis.


Cross Sections that are Squares
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Setting Up the Integral

When you set up the integral, the most important thing to figure out is how to represent the area of each cross section. Think of it as cutting a solid piece of cheese with a thin slice, and think of the function that is generating the area as the face of each slice. The cross section will be perpendicular to either the x or y-axis. If it is perpendicular to the x-axis then your slice is a dx slice and x is your variable for the area function. If it is perpendicular to the y-axis then your slice is a dy slice and y is your variable for the area function. Each slice generates a volume which is found by multiplying the area of the cross section times the width of the slice which is either dx or dy. The integral will be a summation of the volumes created by each slice.


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So let's suppose you want to find the volume of the solid with cross sections that are squares, where the base is the area between the function f(x) = x^2 and the x-axis for 0 ≤ x ≤ 2. The area of each cross section is found using A(x) = (x^2)^2, so the integral to find the volume of the solid will look like this,


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