# Integration, polar coordinates

My question is general rather than specific.If a problem requires to find the area of a figure bounded by a curve given in polar coordinates,how do we find the limits of integration analytically ,without sketching a graph? My problem is mainly with cases that involve trigonometric functions,for example $r^2=2a^2\cos2\theta$.I want to find areas in a purely analytical way without even thinking about the graph of the curve.

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The area element in polar coordinates is $\frac 12 r^2 \; d\theta$ (think of a small wedge of a circle) so you can integrate $\int r(\theta) ^2\frac 12 \; d\theta$

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I think you mean $$\frac{1}{2} \int r(\theta)^2 \, d\theta$$ –  Ron Gordon Apr 3 '13 at 2:47
@RonGordon: Thanks. fixed –  Ross Millikan Apr 3 '13 at 3:33
I am not sure if this constitutes as an answer, but most of time I simply set $r = 0$ and solve and most of the time they will be my bounds.