Area bounded by a Polar Curve

Provided there is a real   r   for every angle between points \begin{align} P(r\,, \alpha) \quad and \quad Q(r\,, \beta) \end{align} the area between the half-lines OP and OQ will be given by

\begin{align} Area & = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta \end{align}

Example

Find the area bounded by the cardioid

\(r = a(1+cos \theta) \)

\begin{align} Area & =\frac{1}{2} \int_{0}^{2 \pi} r^2 d\theta \\ \\ & = \frac{1}{2} \int_{0}^{2 \pi} a^2(1+cos \theta)^2 d\theta \\ \\ & = \frac{1}{2}a^2 \int_{0}^{2 \pi} (cos^2 \theta + 2 \,cos \theta +1) d\theta \\ \\ & = \frac{1}{2}a^2 \int_{0}^{2 \pi} \left( \frac {1}{2}cos2\theta + \frac{1}{2} + 2 \, cos \theta + 1 \right) d\theta \\ \\ & = \frac{1}{2}a^2 \left[ \frac{1}{4}sin2\theta + 2sin \theta + \frac{3}{2} \theta \right]_{0}^{2\pi}\\ \\ & = \frac{1}{2}a^2 \left[ \left( \frac{1}{4}(0) + 2(0) + \frac{3}{2} (2 \pi) \right)- \left(\frac{1}{4}(0) + 2(0) + \frac{3}{2} (0) \right) \right] \\ \\ & = \frac {3 \pi a^2}{2} \end{align}