Chapter 10.3, Problem 45E

To determine

To find: The area of region inside the one petal of four-petaled rose $r=\cos 2\theta$ .

Answer to Problem 45E

The area of region inside the one petal of four-petaled rose is $\frac{\pi }{8}$ .

Explanation of Solution

Given information: The given equation of curve is $r=\cos 2\theta$ .

Formula used: The formula to find the area inside a curve is given by $A=\frac{1}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^{2}d\theta }$ .

Calculation:

Draw the graph of the given equation as shown below.

  

Figure (1)

The area inside of the one petal is equal to the one-fourth of a four petal rose.

To find the area of one petal, integrate the function from $0$ to $2\pi$ .

  $\begin{array}{c}A=\frac{1}{4}\left[\frac{1}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\left(\cos 2\theta \right)}^{2}d\theta }\right]\\ =\frac{1}{8}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{1+\cos 4\theta }{2}d\theta }\\ =\frac{1}{8}\left[\frac{1}{2}{\left.\theta \right|}_0^{2\pi }+\frac{1}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\cos 4\theta d\theta }\right]\\ =\frac{1}{8}\cdot \frac{1}{2}\left[2\pi +\frac{1}{4}\left(0-0\right)\right]\end{array}$

Further simplify to find the area.

  $\begin{array}{c}A=\frac{1}{16}\left[2\pi \right]\\ =\frac{\pi }{8}\end{array}$

Thus, the area of region inside the one petal of four-petaled rose is $\frac{\pi }{8}$ .