# 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=x+2# and #y=x^2# rotated about the x axis?

##### 1 Answer

See the explanation section, below.

#### Explanation:

Graph the region, including the points of intersection,

In order to use shells, we must take our representative slices parallel to the axis of rotation. So the thickness of each shell will be

The thickness of each shell will be

And the limits of integration are

The volume of the representative shell is

The radius of each shell will be

The "height" of each cylindrical shell will be the distance between the

Notice in the picture, that as

**From #y=0# to #y=1#** , we have:

So we need

**From #y=1# to #y=4#** , we have:

So we need

The total volume is