Chapter 10.3, Problem 62E

To determine

To check: Whether the given statement is true or false.

Answer to Problem 62E

The given statement is false.

Explanation of Solution

Given information: The given statement is “The total area enclosed by the three-petaled rose $r=\sin \left(3\theta \right)$ is ${\displaystyle {\int }_0^{2\pi }\frac{1}{2}{\sin }^2\left(3\theta \right)d\theta }$ .”

For a $n$ -petal rose $r=\sin n\theta$ if $n$ is odd, then $\theta \in \left[0,\pi \right]$ .

Consider the given equation of curve $r=\sin \left(3\theta \right)$ . Here, $n=3$ is an odd number.

So, $\theta \in \left[0,\pi \right]$ for a three-petal rose $r=\sin \left(3\theta \right)$ .

To find the area of the three-petal rose integrate the curve from $0$ to $\pi$ .

  $A={\displaystyle {\int }_0^{2\pi }\frac{1}{2}{\sin }^{2}3\theta d\theta }$

The area is not equal to ${\displaystyle {\int }_0^{2\pi }\frac{1}{2}{\sin }^2\left(3\theta \right)d\theta }$ .

Thus, the given statement is false.