Chapter 7.4, Problem 5E

a.

To determine

To find: An integral for the length of the given curve.

Answer to Problem 5E

  $L={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+\frac{1}{2x+2}}dx}$ .

Explanation of Solution

Given:

The curve is $y^2+2y=2x+1$ from $\left(-1,-1\right)$ to $\left(7,3\right)$ .

Concept 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 is,

  $L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx}$

Calculation:

Consider the given curve,

  $y^2+2y=2x+1$

By adding 1 on both sides, it follows that

  $\begin{array}{c}{y}^2+2y+1=2x+1+1\\ {\left(y+1\right)}^2=2x+2\\ y+1=\sqrt{2x+2}\end{array}$

Now differentiating with respect to $x$ , it follows that

  $\begin{array}{c}\frac{dy}{dx}+0=\frac{1}{2\sqrt{2x+2}}\cdot \frac{d}{dx}\left(2x+2\right)\\ =\frac{1}{2\sqrt{2x+2}}\left[2\left(1\right)+0\right]\\ =\frac{1}{\sqrt{2x+2}}\end{array}$

Now the length of the curve from $\left(-1,-1\right)$ to $\left(7,3\right)$ is,

  $\begin{array}{c}L={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx}\\ ={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+{\left[\frac{1}{\sqrt{2x+2}}\right]}^2}dx}\\ ={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+\frac{1}{2x+2}}dx}\end{array}$

b.

To determine

To graph: The given curve.

Answer to Problem 5E

  

Explanation of Solution

Given:

The curve is $y^2+2y=2x+1$ from $\left(-1,-1\right)$ to $\left(7,3\right)$ .

Calculation:

The given curve is a parabola with vertex at $\left(-1,-1\right)$ .

The graph is,

  

c.

To determine

To find: The length of the given curve by using NINT.

Answer to Problem 5E

The length of the curve is about 9.294.

Explanation of Solution

Given:

The curve is $y^2+2y=2x+1$ from $\left(-1,-1\right)$ to $\left(7,3\right)$ .

Calculation:

The integral representing the length of the given curve is $L={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+\frac{1}{2x+2}}dx}$ .

Now by using the NINT, the length of the curve is

  $\begin{array}{c}L={\displaystyle \underset{-1}{\overset{7}{\int }}\sqrt{1+\frac{1}{2x+2}}dx}\\ \approx 9.294\end{array}$