Chapter 5.4, Problem 25P

Find the Jacobians $\partial \left(x,y\right)/\partial \left(u,v\right)$ of the given transformations from variables $x,y$ to variables $u,v$

The volume inside a sphere of radius $r\text{ is }V=\frac{4}{3}\pi r^3$.

Then $dV=4\pi r^{2}dr=Adr$, where A is the area of the sphere. What is the geometrical meaning of the fact that the derivative of the volume is the area? Could you use this fact to find the volume formula given the area formula?