Area of Circle/Proof 2

Theorem

The area $A$ of a circle is given by:

$A = \pi r^2$

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 $2 \pi t \rd t$, since the ring has length $2 \pi t$ and thickness $\rd t$.

 $\displaystyle A$ $=$ $\displaystyle \int_0^r 2 \pi t \rd t$ Perimeter of Circle $\displaystyle$ $=$ $\displaystyle \bigintlimits {\pi t^2} 0 r$ $\displaystyle$ $=$ $\displaystyle \pi r^2$

$\blacksquare$