Chapter 8.4, Problem 6E

a.

To determine

To set up an integral for the length of the curve.

Answer to Problem 6E

The integral for the length of the curve is $L={\displaystyle \underset{0}{\overset{\pi }{\int }}\sqrt{1+{\left(x\sin x\right)}^2}dx}$

Explanation of Solution

Given information:

The given equation is $y=\sin x-x\cos x,\text{ 0}\le x\le \pi$

Formula used:

If a smooth curve begins at $\left(a,c\right)$ and ends at $\left(b,d\right)$ , $a<b,c<d$ , then the length of the curve

  $\begin{array}{l}L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx}\text{ if }y\text{ is smooth function of }x\text{ on }\left[a,b\right]\\ L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}dy}\text{ if }x\text{ is smooth function of }y\text{ on }\left[c,d\right]\end{array}$

Calculation:

Evaluate $\frac{dy}{dx}$

  $\frac{dy}{dx}=x\sin x$

Which is continuous on $\left[0,\pi \right]$ therefore,

  $\begin{array}{l}L={\displaystyle \underset{0}{\overset{\pi }{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx}\\ L={\displaystyle \underset{0}{\overset{\pi }{\int }}\sqrt{1+{\left(x\sin x\right)}^2}dx}\end{array}$

The integral for the length of the curve is $L={\displaystyle \underset{0}{\overset{\pi }{\int }}\sqrt{1+{\left(x\sin x\right)}^2}dx}$

b.

To determine

To graph the curve to see what it looks like.

Explanation of Solution

Given information:

The given equation is $y=\sin x-x\cos x,\text{ 0}\le x\le \pi$

Graph:

  

Interpretation: the part of curve whose length is to be evaluated is shown above.

The crop size is $\left[0,\pi \right]\text{ by }\left[0,4\right]$

c.

To determine

To find the length of curve using NINT.

Answer to Problem 6E

The length of the curve is approximately 4.698

Explanation of Solution

Given information:

The given equation is $y=\sin x-x\cos x,\text{ 0}\le x\le \pi$

Calculation:

First write the integral

  $\begin{array}{l}L={\displaystyle \underset{0}{\overset{\pi }{\int }}\sqrt{1+{\left(x\sin x\right)}^2}dx}\\ =4.698\text{ use NINT}\end{array}$

The length of the curve is approximately 4.698