Q. 31 - Calculus

Question

In exercises 31-36 find a definite integral that represents the length of the specified polar curve, and then use a graphing calculator or computer algebra system to approximate the value of integral

$One petal of the polar rose r=\cos 2\theta$

Step-by-Step Solution

Answer

The integral can be given as ${\int }_0^{2\pi }\sqrt{{(1+\cos 2\theta )}^2+{(-\sin 2\theta )}^2}d\theta$ and the length of the polar curve is  7.43 units

Step 1: Given information

We are given a polar rose of equation $r=\cos 2\theta$

Step 2: Find the integral and evaluate it using computer algebra system

We know that the length of the polar rose can be given as

${\int }_0^{2\pi }\sqrt{(f{\left(\theta \right))}^2+(f\text{'}{\left(\theta \right))}^2}d\theta$

We have

$$\begin{gathered} r=\cos 2\theta \\ r\text{'}=-2\sin 2\theta \end{gathered}$$

Substituting the values we get

${\int }_0^{2\pi }\sqrt{{\left(\cos 2\theta \right)}^2+{(-2\sin 2\theta )}^2}d\theta$

We use computer algebra system to evaluate the integral we get,

$7.43$units