# How to Evaluate the Volume of a Solid of Revolution

You can evaluate the volume of a solid of revolution. A solid of revolution is created by taking a function, or part of a function, and spinning it around an axis — in most cases, either the *x*-axis or the *y*-axis.

*y*= 2 sin

*x*around the

*x*-axis.

For example, the left side of the figure shows the function *y* = 2 sin *x* between *x* = 0 and

Every solid of revolution has circular cross sections perpendicular to the axis of revolution. When the axis of revolution is the *x*-axis (or any other line that’s parallel with the *x*-axis), you can use the meat-slicer method directly.

However, when the axis of revolution is the *y*-axis (or any other line that’s parallel with the *y*-axis), you need to modify the problem.

To find the volume of this solid of revolution, use the meat-slicer method:

Find an expression that represents the area of a random cross section of the solid (in terms of

*x*).This cross section is a circle with a radius of 2 sin

*x**:*Use this expression to build a definite integral (in terms of

*dx*) that represents the volume of the solid.This time, the limits of integration are from 0 to π/2:

Evaluate this integral by using the half-angle formula for sines:

Now evaluate:

So the volume of this solid of revolution is approximately 9.8696 cubic units.