The second term may be considered to be an error term for the first one.
(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?