Volume of revolution

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If we take a curve y = f(x), in xyz space, we can create a 3D shape by rotating the curve about the x-axis. We will use the function:

Here is the function. The RHS shows the function mirrored over the x-axis to show what happens when we rotate it about that axis. The dashed grey ellipses indicate the rotation:

This is called a solid of revolution. We have chosen to create our shape using the section of the curve between x = 2 and x = 5, so we get a solid of finite size. Here is the shape this creates, as a 3D plot:

We can calculate the volume of this 3D shape using integration. This article explains how.

Calculating the volume of revolution for a simple function

We will start by looking at a simple function where the volume can, very easily, be calculated geometrically. We will look at the function y = 2, between x = 1 and x = 4. This function is a horizontal line, parallel to the x-axis. If we rotate it about the x-axis we get a cylinder: