Area of Circle/Proof 4

Theorem

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

$A = \pi r^2$

where $r$ is the radius of the circle.

Proof

Expressing the area in polar coordinates:

 $\displaystyle \iint \rd A$ $=$ $\displaystyle \int_0^r \int_0^{2 \pi} t \rd t \rd \theta$ $\displaystyle$ $=$ $\displaystyle \intlimits {\int_0^r t \theta} 0 {2 \pi} \rd t$ $\displaystyle$ $=$ $\displaystyle \int_0^r 2 \pi t \rd t$ $\displaystyle$ $=$ $\displaystyle 2 \pi \paren {\intlimits {\frac 1 2 t^2} 0 r}$ $\displaystyle$ $=$ $\displaystyle 2 \pi \paren {\frac 1 2 r^2}$ $\displaystyle$ $=$ $\displaystyle \pi r^2$

$\blacksquare$