$\frac{1}{2}{\int }_{{\mathrm{\θ}}_1}^{{\mathrm{\θ}}_2}{f}^2 \mathit{\ⅆ}\mathrm{\θ}-\frac{1}{2}{\int }_{{\mathrm{\θ}}_1}^{{\mathrm{\θ}}_2}{g}^2 \mathit{\ⅆ}\mathrm{\θ}\=\frac{1}{2}{\int }_{{\mathrm{\θ}}_1}^{{\mathrm{\θ}}_2}\left(f^2-g^2\right) \mathit{\ⅆ}\mathrm{\θ}$ 

Example 7.2.1

Working in polar coordinates, calculate the area of the circle $r\=2 a \cos \left(\mathrm{\θ}\right)$.

Example 7.2.2

Working in polar coordinates, calculate the circumference of the circle $r\=2 a \cos \left(\mathrm{\θ}\right)$.

Example 7.2.3

Working in polar coordinates, calculate the area enclosed by the cardioid $r\=1\+ \cos \left(\mathrm{\θ}\right)$.

Example 7.2.4

Working in polar coordinates, calculate the arc length of cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$.

Example 7.2.5

Working in polar coordinates, calculate the complete arc length of the limaçon $r\=1\/2-\cos \left(\mathrm{\θ}\right)$.

Example 7.2.6

Working in polar coordinates, calculate the area within the inner loop of the limaçon $r\=1\/2-\cos \left(\mathrm{\θ}\right)$.

Example 7.2.7

Working in polar coordinates, calculate the area outside the circle $r_1\=2$, but inside the cardioid $r_2\=2\left(1\+\cos \left(\mathrm{\θ}\right)\right)$.

Example 7.2.8

Working in polar coordinates, calculate the area common to the circle $r_1\=\sin \left(\mathrm{\θ}\right)$ and the inner loop of the limaçon $r_2\=1\/5\+\cos \left(\mathrm{\θ}\right)$.

Example 7.2.9

Starting with the expression for the arc length of a curve defined parametrically, obtain the expression for the arc length of a curve defined by $r\=r\left(\mathrm{\θ}\right)$ in polar coordinates. (See Table 7.2.1.)

Example 7.2.10

Obtain $\mathrm{dA}\=\frac{1}{2}{r}^{2}d\mathrm{\θ}$, the area element in polar coordinates.