Solid of revolution - Wikipedia, the free encyclopedia

Best to draw a graph of the region to be revolved.

Suppose integrating with respect to x... mark in a little 'Reimann strip'.

Imagine revolving the strip around the y-axis... yes, a cylindrical shell, so you're integrating with respect to the right variable for shells.

And the expression to integrate will be the circumference 2 pi x times the length of the strip. (4 - x^2 the first time and ln(x) - 0 the second.)

If you've drawn the sketch the limits will be apparent.