# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=2x^2+5#, #y=x+3#, the y-axis, and the line #x=3# rotated about the x-axis?

##### 1 Answer

See the explanation section, below.

#### Explanation:

Here is a graph of part of the region to be rotated about the

In order to use shells, we must take our representative slice parallel to the axis of rotation. In this case, that's the

That means we'll need to rewrite the curves as functions of

Every shell will have

As the graph shows, there are three separate integrals we need to do, because the calculation of

(Are we sure we want to use shells for this?)

**From #y=3# to #y=5#** , the height of the cylindrical shell will be

So we need to integrate

**From #y=5# to #y=6#,** the height of the cylindrical shell will be

So we need to integrate

**From #y=6# to #y=23#,** the height of the cylindrical shell will be

So we need to integrate

**Washers**

To use washers take the slices perpendicular to the axis of rotation.

As x varies from

We need to integrate