Thread: Geometric forumlas related as integrals

I noticed today that the formulas for the area and circumference of a circle are related as integrals. The same goes for the the volume and surface of a cube. In other words:

Also, when I draw the graphs of the two formulas for the circle, they intersect when r=2. With the sphere they intersect at r=3. Does this stem from the fact that the two objects have 2 and 3 dimensions, respectively?

I think I would understand integrals a whole lot better if I knew how this all fits together. So can anyone explain, or at least point me in the right direction?

Thanks

Edit: mods, can you correct my spelling in the title? I obviously meant formulas.

Suppose that V(r) is the ball volume dependent on the radius. Then dV(r) = V'(r)dr is the volume of a thin sphere enveloping the ball whose thickness is dr. On the other hand, the volume of this sphere is S(r)dr where S(r) is the surface area. Therefore, S(r) = V'(r).

Now, suppose that for an $\displaystyle n$-dimensional ball, $\displaystyle V_n(r)=ar^n$ for some constant $\displaystyle a$. Then $\displaystyle V_{n-1}(r)=V_n'(r)=anr^{n-1}$ and the solution to the equation $\displaystyle V_n(r)=V_{n-1}(r)$ is $\displaystyle r = n$.