Q. 31 - Calculus

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. 

Step-by-Step Solution

Answer

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

Step 1: Given information

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

Step 2: 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 $A=\frac{4{\mathrm{\pi }}^3}{6} {\mathrm{unit}}^2$