Chapter A6, Problem 64E

(a)

To determine

To find: the length of the curve.

Answer to Problem 64E

The length of the curve is $\frac{6}{5}$

Explanation of Solution

Given:

  $y=\left(\frac{1}{2}\right)\cosh 2x,0\le x\le \text{ 1}n\sqrt{5}$

Calculation:

find the length of curve from $x=atox=b$ we just have to find the integral,

  ${\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{{\left(1+y\text{'}\right)}^2}\cdot dx}$

Therefore the integral to be calculated is,

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{1n\sqrt{5}}{\int }}\sqrt{1+{\left(\frac{\frac{d}{dx}\cosh 2x}{2}\right)}^2}dx=}{\displaystyle \underset{0}{\overset{1n\sqrt{5}}{\int }}\sqrt{1+{\left(\sinh 2x\right)}^{2}dx}}\\ ={\displaystyle \underset{0}{\overset{1n\sqrt{5}}{\int }}\cosh 2xdx}\\ ={\left[\frac{1}{2}\sinh 2x\right]}_0^{1n\sqrt{5}}\\ ={\left[\frac{1}{2}\left(\frac{e^{2x}-e^{-2x}}{2}\right)\right]}_0^{1n\sqrt{5}}\\ =\frac{1}{4}\left(5-\frac{1}{5}\right)\\ =\frac{6}{5}\end{array}$

Conclusion:

The length of the curve is $\frac{6}{5}$

(b)

To determine

To find: the length of the curve.

Answer to Problem 64E

The length of the curve is $\frac{\sinh ab}{a}$

Explanation of Solution

Given:

  $y=\left(\frac{1}{a}\right)\cosh ax,0\le x\le \text{b}$

Calculation:

find the length of curve from $x=atox=b$ , we just have to find the integral,

  ${\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{{\left(1+y\text{'}\right)}^2}\cdot dx}$

Therefore the integral to be calculated is,

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{b}{\int }}\sqrt{1+{\left(\frac{\frac{d}{dx}\mathrm{cosha}x}{a}\right)}^2}\cdot dx=}{\displaystyle \underset{0}{\overset{b}{\int }}\sqrt{1+{\sinh }^{2}ax\cdot dx}}\\ ={\displaystyle \underset{0}{\overset{b}{\int }}\cosh ax\cdot dx}\\ ={\left[\frac{1}{a}\sinh ax\right]}_0^b\\ =\frac{\sinh ab}{a}\end{array}$

Conclusion:

The length of the curve is $\frac{\sinh ab}{a}$