Area of Circle/Proof 2

Theorem
The area $A$ of a circle is given by the formula $A=\pi r^2$, where $r$ is the radius of the circle.

Proof
Proof by shell integration:

The circle can be divided into a set of infinitesimally thin rings, each of which has area $2 \pi t \ \mathrm dt$, since the ring has length $2 \pi t$ and thickness $\mathrm d t$.