# Volumes of solid of revolution - Shell method

### Volumes of solid of revolution - Shell method

In this section, we will learn that we cannot always find the volume of a solid using the disc method. In fact we see that the disc method fails for certain functions, which Desmond demonstrates in the intro video. So instead of integrating the area of circles, we will integrate the area of cylindrical shells. This is known as the shell method. This is also important because we know the formula for the area of a cylinder, hence we can integrate it. We will be looking at questions which involve rotating around not only the x-axis, but the y-axis and different x values.

#### Lessons

• Introduction

• 1.
Find the volume of the solid obtained by rotating the region bounded by the x-axis and $y=3x-x^4$ about the y-axis

• 2.
Find the volume of the solid obtained by rotating the positive region bounded by $y=x^3$, $y=x$
a)
about $x=2$
about $x=-2$
Use the Shell Method to find the volume of the solid obtained by rotating about the x-axis the region under $y=x^3$ from $x=0$ to $x=2$