Question

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one loop of $r^2=4\cos 2\theta$.

Short answer

The area bounded by one loop is$2\mathrm{units}$

Step-by-step solution

Given information

$r^2=4\cos 2\theta$

Area

The graph of the given function is,

The area bounded by one loop is,

$$\begin{gathered} A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta \\ A=\frac{1}{2}{\int }_{-\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}4\cos 2\theta d\theta \\ A=2{\left(\frac{\sin 2\theta }{2}\right)}_{-\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}} \\ A=2\left(\frac{\sin 2\left(\frac{\mathrm{\pi }}{4}\right)}{2}-\frac{\sin 2\left(-\frac{\mathrm{\pi }}{4}\right)}{2}\right) \\ A=4\left(\frac{1}{2}+\frac{1}{2}\right) \\ A=2\mathrm{units} \end{gathered}$$

The required area is$A=2\mathrm{units}$