Question

Prove that the part of the polar curve $r=\frac{1}{\theta }$that lies inside the circle defined by the polar equation r = 1 has infinite length.

Short answer

It is proved that the part of the polar curve that lies inside the circle defined by the polar equation r = 1 has infinite length.

Step-by-step solution

Step 1. Given information

An equation of a polar curve is$r=\frac{1}{\theta }$

Step 2. Explanation

Length $={\int }_1^{\infty }\sqrt{\frac{1}{{\theta }^4}+\frac{1}{{\theta }^2}}d\theta$

$$\begin{gathered} ={\int }_1^{\infty }\sqrt{\frac{1+{\theta }^2}{{\theta }^4}}d\theta \\ ={\left[\sin h^{-1}\theta -\frac{\sqrt{1+{\theta }^2}}{\theta }\right]}_1^{\infty } \\ =\left[\sin h^{-1}\infty -\frac{\sqrt{1+{\infty }^2}}{\infty }-\sin h^{-1}1+\frac{\sqrt{1+1^2}}{1}\right] \\ =\infty \end{gathered}$$