Chapter 10.3, Problem 4QQ

i.

To determine

To calculate: The area bounded by the curve.

Answer to Problem 4QQ

The Area bounded by the curve is $4.382$ .

Explanation of Solution

Given information: The given curve as follows:

The polar curve is defined by the equation $r=\theta +\sin 2\theta$ for $0\le \theta \le \pi$ .

Calculation:

Since , the area can be calculated as follows:

Radius is given as $r=\theta +\sin 2\theta$ .

Area of the curve is $=\frac{1}{2}{\displaystyle {\int }_a^b{r}^{2}d\theta }$ .

  $\begin{array}{c}\text{A}=\frac{1}{2}{{\displaystyle {\int }_0^{\pi }\left(\theta +\sin 2\theta \right)}}^{2}d\theta \\ \text{A}=\frac{1}{2}{\displaystyle {\int }_0^{\pi }\left({\theta }^2+{\sin }^{2}2\theta +2\theta \sin 2\theta \right)}d\theta \\ \text{A}=\frac{1}{2}{\displaystyle {\int }_0^{\pi }\left(\frac{{\theta }^3}{3}+\frac{1-\cos 2\theta }{2}+2\theta \sin 2\theta \right)}d\theta \\ \text{A}=\frac{1}{2}{\displaystyle {\int }_0^{\pi }\left(\frac{{\theta }^3}{3}+\frac{\theta }{2}-\frac{\sin 2\theta }{4}+2\left[-\theta \frac{\cos 2\theta }{2}+\frac{\sin 2\theta }{4}\right]\right)}\\ \text{A}=\frac{1}{2}{\left(\frac{{\theta }^3}{3}+\frac{\theta }{2}-\frac{\sin 2\theta }{4}+2\left[-\theta \frac{\cos 2\theta }{2}+\frac{\sin 2\theta }{4}\right]\right)}_0^{\pi }\\ A=\frac{1}{2}\left(\frac{{\pi }^3}{3}+\frac{\pi }{2}-\frac{\sin 2\pi }{4}+2\left[-\pi \frac{\cos 2\pi }{2}+\frac{\sin 2\pi }{4}\right]-0\right)\\ A=4.382\end{array}$ .

ii.

To determine

To calculate: To solve the value of the angle $\theta$ .

Answer to Problem 4QQ

The value of the angle $\theta =2.786$ .

Explanation of Solution

Given information: The given curve as follows:

The polar curve is defined by the equation $r=\theta +\sin 2\theta$ for $0\le \theta \le \pi$ . The point on the curve that corresponds with the $x-$ coordinate is $-2$ .

Calculation:

The value can be calculated as:

  $\begin{array}{c}x=r\cos \theta \\ r=\theta +\sin 2\theta \\ -2=\left(\theta +\sin 2\theta \right)\cos \theta \\ -2=\theta \cos \theta +2\sin \theta {\cos }^2\theta \\ 2\sin \theta {\cos }^2\theta +\theta \cos \theta +2=0\\ \theta =2.786\end{array}$

iii.

To determine

To calculate: The facts about the curve.

Answer to Problem 4QQ

The curve is getting closer to the origin.

Explanation of Solution

Given information: The given curve as follows:

The polar curve is defined by the equation $r=\theta +\sin 2\theta$ for $0\le \theta \le \pi$ . For $\frac{\pi }{3}<\theta <\frac{2\pi }{3}$ , $\frac{dr}{d\theta }$ is negative.

Calculation:

The curve can be defined as follows:

Since, $\frac{dr}{d\theta }<0$ for $\frac{\pi }{3}<\theta <\frac{2\pi }{3}$ , $r$ is decreasing on this interval. This means the curve is getting closer to the origin.

iv.

To determine

To calculate: The value of $\theta$ .

Answer to Problem 4QQ

The value of the angle $\theta =\frac{\pi }{3}$ .

Explanation of Solution

Given information: The given curve as follows:

The polar curve is defined by the equation $r=\theta +\sin 2\theta$ for $0\le \theta \le \pi$ . For $0\le \theta \le \frac{\pi }{2}$ , that corresponds to the point to the curve in the first quadrant with greatest distance from the origin.

Calculation:

Since, the curve can be calculated as:

  $\begin{array}{c}\frac{dr}{d\theta }=1+\cos 2\theta \\ \frac{dr}{d\theta }=0\end{array}$

Is $\theta =\frac{\pi }{3}$ .

The table can be calculated as:

  

The greatest distance occurs when $\theta =\frac{\pi }{3}$ .