# Volumes by Cylindrical Shells Method

Let’s consider the problem of finding the volume of the solid obtained by rotating about the $x$-axis or parallel to $x$-axis the region, where the core idea of cylindrical shells method for finding volumes. If we slice perpendicular to the $y$-axis, we get a cylinder. But to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation for $x$ in terms of $y$. The method of cylindrical shells is being used for finding the volume in this case, that is easier to use in such a case. We can see a cylindrical shell with inner radius, outer radius, and height . Its volume is calculated by subtracting the volume of the inner cylinder from the volume of the outer cylinder.

### Worked Example of Finding a Volume by Cylindrical Shells Method

The diagram shows the graph of $\displaystyle f(x) = \frac{x}{1 + x^2}$. The area bounded by $y = f(x)$, the line $x = 1$ and the $x$-axis is rotated about the line $x=1$ to form a solid volume. Use the cylindrical shells method to find the volume of the solid. 