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

##### 1 Answer

See explanation below.

#### Explanation:

To use shells when revoling about a vertical line, we need to take our representative slices vertically, so the thickness is

Here is a picture of the region to be revolved about the

The curves intersect at

The radii of our cylindrical shells will be

The thickness is

the height of each shell is the greater

(The

The greater value of

The lesser value of

**From #x=0# to #x=1#,** the lesser

To get that part of the volume, we need to evaluate:

**From #x=1# to #x=4#,** the lesser

To get that part of the volume, we need to evaluate:

**The total volume** is found by adding these two results, to get: