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Chapter 9: Q 69. (page 723)

Use your result from Exercise $68$ to show that the arc length formula for a function $y = f (x)$ is a special case of the arc length formula for a parametric curve.

Short Answer

Expert verified

${鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t$

Step by step solution

01

Given information

$y = f (x)$

02

Calculation

Consider the function $y = f (x)$

For a parameter $t$ the function $y = f (t)$ for some $t 鈭� [a, b]$

The goal is to determine the curve's arc length.

If the curve $C$ is expressed by parametric equations $x = f (t), y = y (t)$ on the interval $[a, b]$ then the arc length is given by the formula,

${鈭�}_{a}^{b} \sqrt{{(f^{'} (t))}^{2} + {(g^{'} (t))}^{2}} d t$

Thus, $f (t) = x 鈬� f^{'} (t) = \frac{d x}{d t}$

$g (t) = y 鈬� g^{'} (t) = \frac{d y}{d t}$

Substituting the values of $f^{'} (t), g^{'} (t)$ then the arc length is

Arc length $= {鈭�}_{a}^{b} \sqrt{{(\frac{d}{d t} x)}^{2} + {(\frac{d}{d t} y)}^{2}} d t$

$= {鈭�}_{a}^{b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$

Then,

Arc length $= {鈭�}_{a}^{b} \sqrt{1 + {(\frac{d y}{d t})}^{2}} d t$

Arc length $= {鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t [$ since , for some parameter $t]$ Therefore the arc length of the curve $y = f (t)$ is ${鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (t))}^{2}} d t$

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Most popular questions from this chapter

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

The limacon $r = 2 + \cos 胃$

Prove that for an ellipse or a hyperbola the eccentricity is given by

$e = \frac{the distance between the foci}{the distance between the vertices}$

Sketch the graphs of the equations

$r = \frac{2}{2 + \cos 胃}$ and localid="1649860998050" $r = \frac{2}{2 + \sin 胃}$

What is the relationship between these graphs? What is the eccentricity of each graph?

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$胃 = 蟺$

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

$(0, - 3)$

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