Question

Areas of regions bounded by polar functions: Find the areas of the following regions.

The area bounded by the function $r=a\cos \theta$, where $a$ is a positive constant.

Short answer

The area bounded by the function is$\frac{{\mathrm{\pi a}}^2}{2}{\mathrm{unit}}^2$

Step-by-step solution

Given information.

The given function is$r=a\cos \theta$,where a is a positive constant.

Area

The area of the curve is given by,

$$\begin{gathered} A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{2\mathrm{\pi }}{\left(a\cos \theta \right)}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{2\mathrm{\pi }}\left(a^2{\cos }^2\theta \right)d\theta \\ A=\frac{a^2}{2}{\int }_0^{2\mathrm{\pi }}\left(\frac{1+\cos 2\theta }{2}\right)d\theta \\ A=\frac{a^2}{4}{\int }_0^{2\mathrm{\pi }}\left(\frac{1+\cos 2\theta }{2}\right)d\theta \\ A=\frac{a^2}{4}{\left[\theta +\frac{\sin 2\theta }{2}\right]}_0^{2\mathrm{\pi }} \\ AA=\frac{a^2}{4}{\left[2\mathrm{\pi }+\frac{\sin 4\mathrm{\pi }}{2}\right]}_0^{2\mathrm{\pi }} \\ A=\frac{{\mathrm{\pi a}}^2}{2}{\mathrm{unit}}^2 \end{gathered}$$

The area of the function is$A=\frac{{\mathrm{\pi a}}^2}{2}{\mathrm{unit}}^2$