Chapter 10.1, Problem 34E

To determine

To find: The length of the curve.

Answer to Problem 34E

The answer:

  $L=\ln 2$

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=\ln (\sec t+\tan t)-\sin t,\\ y=\cos t,0\le t\le \pi /3\end{array}$

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,

  $\begin{array}{c}x=\ln (\sec t+\tan t)-\sin t,\\ y=\cos t,\end{array}$

And

  $0\le t\le \pi /3$

Know that,

  $\begin{array}{c}\sec t=1/\cos t\\ \tan t=\sin t/\cos t\end{array}$

Then,

  $\begin{array}{c}{t}_1=0,\\ t_2=\pi /3,\\ dy/dt=-\sin t\end{array}$

And,

  $\begin{array}{c}\frac{dx}{dt}=\frac{1}{\sec t+\tan t}\cdot \left(-\frac{1}{{\cos }^{2}t}\cdot (-\sin t)+\frac{\cos t\cdot \cos t-\sin t\cdot (-\sin t)}{{\cos }^{2}t}\right)-\cos t\\ =\frac{1}{\sec t+\tan t}\cdot \left(\frac{\sin t}{{\cos }^{2}t}+\frac{{\cos }^{2}t+{\sin }^{2}t}{{\cos }^{2}t}\right)-\cos t\\ =\frac{1}{\sec t+\tan t}\cdot \left(\frac{1}{\cos t}\cdot \frac{\sin t}{\cos t}+\frac{1}{{\cos }^{2}t}\right)-\cos t\\ =\frac{\sec t\tan t+{\sec }^{2}t}{\sec t+\tan t}-\cos t\\ =\frac{\sec t\cdot (\tan t+\sec t)}{\sec t+\tan t}-\cos t\\ =\sec t-\cos t\end{array}$

The foregoing findings lead to the conclusion that the parametric curve is traced only once for each $t$ between $0$ and $\pi /3$ that $dx/dt$ and $dy/dt$ are continuous functions of $t$ for $0\le t\le \pi /3$. utilise the above findings to determine that the length of the parametric curve is equal to:

  

To solve the integral,

  

Use the substitution,

  $u=\cos t$

Then,

  $du=-\sin tdt$

For,

  $t=0$

Then,

  $\begin{array}{c}u=\cos 0\\ =1\end{array}$

While for,

  $t=\pi /3$

Then,

  $\begin{array}{c}u=\cos (\pi /3)\\ =1/2\end{array}$

To use the previous result,

  

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

  

Therefore, the required length of the curve is $L=\ln 2$ .