# The area bounded by the curve #y=3+2x-x^2# and line #y=3# is rotated completely about the line #y=3#. Find the volume of the solid of revolution obtained?

##### 1 Answer

#### Answer:

#### Explanation:

The area are the solution of this system:

And it is sketched in this plot:

The formula for the volume of a x-axis rotation solid is:

To apply the formula we should translate the half moon on the x-axis, the area won't change, and so it won't change also the volume:

In this way we obtain

The translated area now is plotted here:

But which are the a and b of the integral? The solutions of the system:

So

Let's rewrite and solve the integral:

And this "lemon" is the solid obtained: