Q. 47 - Calculus

Question

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r=a$ is $2\pi a$.

Step-by-Step Solution

Answer

The length of the curve is $\begin{array}{rcl}& & 2\pi a\end{array}$.

Step 1. Given Information

The function that bounds the circle is $r=a$.

Step 2. Use the formula of arc length of a polar curve

• The region is bounded by the curve $r=a$.
• The region is a circle, so the boundary arcs will be $\theta =0 \mathrm{to} \theta \mathit{=}\mathit{2}\pi$.
• So, the length of the curve will be calculated as follows: 

$\begin{array}{rcl}{\int }_0^{2\mathrm{\pi }}\sqrt{{(r\text{'})}^2+r^2}d\theta & =& {\int }_0^{2\mathrm{\pi }}\sqrt{0+a^2}d\theta \\ & =& a{\int }_0^{2\mathrm{\pi }}d\theta \\ & =& a\times 2\mathrm{\pi }\\ & =& 2\pi a\end{array}$

• Hence, the circumference of the curve is $\begin{array}{rcl}& & 2\pi a\end{array}$.