# 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.