Chapter 11.2, Problem 10QR

To determine

To find: The length of the curve that is defined parametrically by $x=\sin \left(2t\right)$ and $y=\cos 3t$ for $0\le t\le 2\pi$ .

Answer to Problem 10QR

The length of the curve is $15.294$ .

Explanation of Solution

The derivative of the both the equation with respect to $x$ is,

  $\begin{array}{l}\frac{dx}{dt}=2\cos 2t\\ \frac{dy}{dt}=-3\sin \left(3t\right)\end{array}$

Then, the length of the parametric curve is defined by,

  $L={\displaystyle {\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$

Then,

  $\begin{array}{c}L={\displaystyle {\int }_0^{2\pi }\sqrt{\left[2\cos {\left(2t\right)}^2\right]+\left[-3\sin {\left(3t\right)}^2\right]}}dt\\ =15.294\end{array}$

Thus, the length of the curve is $15.294$ .