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 $\int r(\theta) ^2\frac 12 \; d\theta$
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.
What is so not obvious is if we are given two or more polar curves and are asked to find the area of their intersections.