(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height

, inner radius

, and thickness

.

This part is fairly simple--

, assuming

is a constant. This yields

.

(b) What is the error involved in using the formula from part (a)?

This is where I'm stuck. The formula in part (a) assumes

is a constant, and thus doesn't account for the change in

, even though the thickness does affect the top of the shell. Seeing this, I would assume the error to be in the volume of the "lid", or

. However, my book gives me an answer of

. What am i doing wrong?