$A\=\frac{1}{2}{\int }_{{\mathrm{\θ}}_1}^{{\mathrm{\θ}}_2}\left(r_2^2-r_1^2\right) \mathit{\ⅆ}\mathrm{\θ}$ 

where ${\mathrm{\θ}}_1\=-\mathrm{\π}\/2$ and ${\mathrm{\θ}}_2\=\mathrm{\π}\/2$. The animation in Figure 7.2.3(a) shows that the first-quadrant portion of the cardioid is traced for $\mathrm{\θ}\in \left[0\,\mathrm{\π}\/2\right]$. The animation in Figure 7.2.7(a) shows the polar element of area sweeping the region $r_1\le r\le r_2$ with $\mathrm{\θ}\in \left[-\mathrm{\π}\/2\,\mathrm{\π}\/2\right]$.

$\frac{1}{2}{\int }_{-\mathrm{\π}\/2}^{\mathrm{\π}\/2}\left({\left(2\left(1\+\cos \left(\mathrm{\θ}\right)\right)\right)}^2-2^2\right) \mathit{\ⅆ}\mathrm{\θ}$

$\=2{\int }_{-\mathrm{\π}\/2}^{\mathrm{\π}\/2}\left(2 \cos \left(\mathrm{\θ}\right)\+{\cos }^2\left(\mathrm{\θ}\right)\right) \mathit{\ⅆ}\mathrm{\θ}$

$\=2{\int }_{-\mathrm{\π}\/2}^{\mathrm{\π}\/2}\left(2 \cos \left(\mathrm{\θ}\right)\+\frac{1\+\cos \left(2 \mathrm{\θ}\right)}{2}\right) \mathit{\ⅆ}\mathrm{\θ}$

$\=2{\left[2 \sin \left(\mathrm{\θ}\right)\+\frac{\mathrm{\θ}}{2}\+\frac{\sin \left(2 \mathrm{\θ}\right)}{4}\right]}_{-\mathrm{\π}\/2}^{\mathrm{\π}\/2}$

$\=2\left(4\+\frac{\mathrm{\π}}{2}\+0\right)$

$\=\mathrm{\π}\+8$

Interactive solution

$\frac{1}{2}{\int }_{-\mathrm{\pi }\/2}^{\mathrm{\pi }\/2}\left({\left(2\left(1\+\cos \left(\mathrm{\θ}\right)\right)\right)}^2-2^2\right) \mathit{\ⅆ}\mathrm{\θ}$ = $\mathrm{\π}\+8$$$