Examples

Circle with polar equation r = 2cos(theta)

The points whose polar coordinates satisfy $r=2\cos (\theta )$ define a circle:

We can verify this by converting to Cartesian coordinates:

$$x=r\cos (\theta )=\frac{1}{2}{r}^2=\frac{1}{2}(x^2+y^2),$$

so

$$(x-1{)}^2+y^2=1.$$

Area of one loop of a four-leafed rose

Find the area of one loop of the four-leafed rose $r=\cos (2\theta )$.

Solution: Plug into the formula, and use $a=-\frac{\pi }{4},b=\frac{\pi }{4}$, obtaining:

$$A={\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{1}{4}(1+\cos (4\theta ))d\theta =\frac{\pi }{8}.$$