Area of Circle/Proof 4

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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:

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

$\blacksquare$