Chapter 11.1, Problem 30E

To determine

To Find:The length of the curve.

Answer to Problem 30E

Therequired length of the curve is $12$

Explanation of Solution

Given Information:

  $x=2{\cos }^{3}t$  ........ (1)

  $y=2{\sin }^{3}t$  ........ (2)

  $0\le t\le 2\pi$

Formula Used:

Length of a curve is given by

  $L={\displaystyle \underset{t1}{\overset{t2}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$  ........ (3)

Calculation:

Differentiating equation (1) and (2) with respect to $t$

  $\frac{dx}{dt}=-6\sin t{\cos }^{2}t$  ........ (4)

  $\frac{dy}{dt}=6{\sin }^{2}t\cos t$  ........ (5)

Substitute $-6\sin t{\cos }^{2}t$ for $\frac{dx}{dt}$ , and $6{\sin }^{2}t\cos t$ for $\frac{dy}{dt}$ in equation (3)

  $\begin{array}{c}L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt\\ ={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{{\left(-6\sin t{\cos }^{2}t\right)}^2+{\left(6{\sin }^{2}t\cos t\right)}^2}}dt\\ ={\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(6\sin t\cos t\right)}dt\\ =12\end{array}$

Conclusion:

Hence, the required length of the curve is $12$