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 yaxis. 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, 2πx. So to find the area of a shell, multiply these two numbers together:
A = 2πx cos 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 edge

Evaluate 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.