Q. 23 - Calculus

Question

The arc length of polar functions: Find the arc lengths of the following polar functions. 

$r=a$, where $a$ is a positive constant, for $\theta \in [0, 2\pi ]$

Step-by-Step Solution

Answer

The arc length is $2a\mathrm{\pi } \mathrm{units}$

Step 1: Given information.

The curve $r=a$ and $\theta \in \left[0, 2\mathrm{\pi }\right]$.

Step 2: Length of an arc.

The length of an arc is given by:

$L={\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta$

So,

$$\begin{gathered} L={\int }_0^{2\mathrm{\pi }}\sqrt{a^2+0^2}d\theta \left[\because \frac{da}{d\theta }=0\right] \\ ={\int }_0^{2\mathrm{\pi }}\sqrt{a^2}d\theta \\ ={\int }_0^{2\mathrm{\pi }}a d\theta \\ =2a\mathrm{\pi }-0 \\ =2\mathrm{a\pi } \end{gathered}$$

Step 3: Conclusion.

The arc length of the given polar is $2a\mathrm{\pi } \mathrm{units}$.