11.5 - Ice cream and the chain rule

Consider the solid obtained when rotating the region (right) about the $y$ axis. The volume of the solid is approximately the volume of ice cream needed to fill and top an ice-cream cone with the dimensions shown.

1. Compute the volume of this solid. (The volume of a cone, though you *can* compute it using calculus, is $V_c=\pi r^2 h/3$.)
2. Find the volume $V(h,r)$ of a similar solid created by rotating a region with dimensions $h$ and $r$ instead of 3 and 2.
3. What's the meaning of $\frac{\del V}{\del r}$?
4. Suppose that $h$ and $r$ are varying with time according to $h(t)=t+\sin t$ and $r(t)=e^t-\cos t$. Compute $dV/dt$.