[tex]Area\;of\;circle:\\\\A=\pi r^2\\\\\\proof\\\\A= \int\int\limits_D \, dxdy\\\\ D:x^2+y^2\le r\\\\ \left \{ {{y=Rsin\phi} \atop {x=Rcos\phi}} \right. \\\\J=R\\\\R\in[0;r]\\\\\phi\in[0;2\pi]\\\\A= \int\limits^{2\pi}_0 \, d\phi \int\limits^r_0 {R} \, dR = \frac{1}{2} r^2\int\limits^{2\pi}_0 \, d\phi=\pi r^2[/tex]
