Chapter 10.1, Problem 28E

To determine

To find: The length of the curve.

Answer to Problem 28E

The answer:

  $L=3\pi$

Explanation of Solution

Given information:

The equations are:

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

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,

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

And

  $0\le t\le \pi$

Then,

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

And

  $\begin{array}{c}\frac{dx}{dt}=3\cos t\\ \frac{dy}{dt}=-3\sin t\end{array}$

The foregoing findings lead to the conclusion that the parametric curve is traced only once for each $t$ between $0$ and $\pi$ that $dx/dt$ and $dy/dt$ are continuous functions of $t$ for $0\le t\le \pi$ . 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:

  

Therefore, the required length of the curve is $3\pi$ .