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

This needs considerable tedious hard slog to complete it. In particular: Add the required link to the demonstration that the small unit of area is $t \rd t \rd \theta$ To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Expressing the area in polar coordinates:

\(\ds \iint \rd A\)

\(=\)

\(\ds \int_0^r \int_0^{2 \pi} t \rd t \rd \theta\)

\(\ds \)

\(=\)

\(\ds \intlimits {\int_0^r t \theta} 0 {2 \pi} \rd t\)

\(\ds \)

\(=\)

\(\ds \int_0^r 2 \pi t \rd t\)

\(\ds \)

\(=\)

\(\ds 2 \pi \paren {\intlimits {\frac 1 2 t^2} 0 r}\)

\(\ds \)

\(=\)

\(\ds 2 \pi \paren {\frac 1 2 r^2}\)

\(\ds \)

\(=\)

\(\ds \pi r^2\)

$\blacksquare$