Find the volume of the region bounded by the following equations?

I have drawn a graph of this problem and put most of the early steps on that graph so that you can see the graph and the work in one window. To see the graph (necessary) click on the following link:

http://i369.photobucket.com/albums/oo133/gerryrain...

This is a repeat of the steps on the graph.

First we have to find where x = 3y^2 intersects y = 3. Since 3 * (3^2) =

3 * 9 = 27, the point of intersection is (27,3).

We're going to use the method of cylindrical shells.

The "height" is 3y^2. The radius is (3 - y)

So we want to integrate 2 *π *(3-y) * (3y^2) = 18πy^2 - 6πy^3 dy for 0 < y < 3

3

∫ 6π * (3y^2 - y^3) dy = 6π * (y^3 - (y^4 / 4)) as y goes from 0 to 3

0

At 0 this evaluates to 0.

At 3 this evaluates to 6π * (27 - 81/4) = 6π * ( (108 / 4) - (81 / 4)) = 6π * (27/4)

= 81π/2 <------------ Answer

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