Question

How do you find the volume of the solid formed by revolving a particular region around the x-axis given `y=2`, `y=4-(x^2/2)` and bounded from [-2,2]?

Answer 1

This is a very good question indeed!!!!
First have a look at the region on the `xy` axis that will rotate:

Basically when this region rotates about the `x` axis it will describe a kind of "fat" ring.
To evaluate the volume you can first rotate the curved part alone and get its volume (a barrel) and then subtract the volume obtained rotating the straight line alone (a cylinder) as in:

To evaluate each volume I use a small cylinder of radius `y` (given by the function itself) and thickness `deltax` that gives you a small volume `deltaV`. The complete volume `V` will be obtained "integrating" the small cylinder along the `x` axis:

So that:

And finally:

Hope it helps but PLEASE check my maths!