Q. 33 - Calculus

Question

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

One petal of the polar rose $r=\cos 4\theta$

Step-by-Step Solution

Answer

The integral can be given as $2{\int }_0^{\frac{\pi }{8}}\sqrt{{\cos }^{2}4\theta +16{\sin }^{2}4\theta }d\theta$ and the arc length can be given as 2.14units.

Step 1: Given information

We are given the equation as $r=\cos 4\theta$

Step 2: Evaluate

We know that the arc length of one petal of the polar rose $r=\cos \left(n\theta \right)$ can be given as

$$\begin{gathered} 2{\int }_0^{\frac{\pi }{2n}}\sqrt{{\cos }^{2}n\theta +n^2{\sin }^2\theta }d\theta \\ Put n=4 we get, \\ 2{\int }_0^{\frac{\pi }{8}}\sqrt{{\cos }^{2}4\theta +16{\sin }^{2}4\theta }d\theta \end{gathered}$$

Using a CAS calculator we get,

$$\begin{gathered} L=2(1.07) \\ L=2.14unit \end{gathered}$$