Question

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by the function $r=a$, where $a$ is a positive constant.

Short answer

The area bounded by the function is$\frac{4{\mathrm{\pi }}^3}{6}{\mathrm{unit}}^2$

Step-by-step solution

Given information

The function is$r=a$, where$a$is a positive integer.

Area 

The area of the curve is given by,

$$\begin{gathered} A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{2\mathrm{\pi }}{a}^{2}d\theta \\ A=\frac{1}{2}{\left[\frac{a^3}{3}\right]}_0^{2\mathrm{\pi }} \\ A=\left[\frac{{\left(2\mathrm{\pi }\right)}^3}{6}-0\right] \\ A=\left[\frac{8{\mathrm{\pi }}^3}{6}\right] \\ A=\frac{4{\mathrm{\pi }}^3}{6}{\mathrm{unit}}^2 \end{gathered}$$

The area of the curve is$A=\frac{4{\mathrm{\pi }}^3}{6}{\mathrm{unit}}^2$