# Using the Shell Method to Find the Volume of a Solid of Revolution

The shell method is useful when you’re measuring a volume of revolution around the *y*-axis. For example, suppose that you want to measure the volume of the solid shown in this figure.

Here’s how the shell method can give you a solution:

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

*x*).Remember that each shell is a rectangle with two different sides: One side is the height of the function at

*x*— that is, cos*x*. The other is the circumference of the solid at*x*— that is, 2*π**x.*So to find the area of a shell, multiply these two numbers together:*A*= 2*π**x**x*Use this expression to build a definite integral (in terms of

*dx*) that represents the volume of the solid.In this case, remember that you’re adding up all the shells from the center (at

*x*= 0) to the outer edgeEvaluate the integral.

This integral is pretty easy to solve using integration by parts:

Now evaluate this expression:

So the volume of the solid is approximately 0.5708 cubic units.