Chapter 10.1, Problem 30E

To determine

To find: The length of the curve.

Answer to Problem 30E

The answer:

  $L=12$

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=2{\cos }^{3}t,\\ y=2{\sin }^{3}t,0\le t\le 2\pi \end{array}$

Calculation:

Know that,

If $dx/dt$

And

  $dy/dt$ are continuous functions of $t$ for $t_1\le t\le t_2$ .then the length of a parametric curve that is traced only once for each $t$ between $t_1$ and $t_2$ equal to:

  

Since,

  $\begin{array}{c}x=2{\cos }^{3}t,\\ y=2{\sin }^{3}t\text{,}\end{array}$

And

  $0\le t\le 2\pi$

Then,

  $\begin{array}{c}{t}_1=0\\ t_2=2\pi \end{array}$

And,

  $\begin{array}{c}\frac{dx}{dt}=2\cdot 3{\cos }^{2}t\cdot (-\sin t)\\ =-6{\cos }^{2}t\sin t\\ \frac{dy}{dt}=2\cdot 3{\sin }^{2}t\cdot \cos t\\ =6{\sin }^{2}t\cos t\end{array}$

The foregoing findings lead to the conclusion that the parametric curve is traced only once for each $t$ between $0$ and $2\pi$ that $dx/dt$ and $dy/dt$ are continuous functions of $t$ for $0\le t\le 2\pi$ . Due to the fact that ${\sin }^{2}t+{\cos }^{2}t=1$ for each value of $t$, utilise the above findings to determine that the length of the parametric curve is equal to:

  

To solve the integral,

  

use the substitution

  $u=\sin t$

Then,

  $du=\cos tdt$

For,

  $t=0$

So,

  $\begin{array}{c}u=\sin (0)\\ =0\end{array}$

While for

  $t=\pi /2$

So,

  $\begin{array}{c}u=\sin (\pi /2)\\ =1\end{array}$

To use the previous result,

  

The previous find indicate that the parametric curve's length is equal to:

  

Therefore, the required length of the curve is $L=12$ .