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**Chokfull** (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height $\displaystyle h$, inner radius $\displaystyle r$, and thickness $\displaystyle \Delta r$.

This part is fairly simple--$\displaystyle dV=f'(r)*dr$, assuming $\displaystyle h$ is a constant. This yields $\displaystyle dV=2\pi rh\Delta r$.

(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 $\displaystyle h$ is a constant, and thus doesn't account for the change in $\displaystyle h$, 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 $\displaystyle \pi r^2 \Delta r$. However, my book gives me an answer of $\displaystyle \pi (\Delta r)^2 h$. What am i doing wrong?