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Chapter 9: Q. 26 (page 775)

The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = \sin 胃 + \cos 胃 for 胃鈭� [0, 蟺]$

Short Answer

Expert verified

The arc length is $2 \sqrt{2} 蟺 units$

Step by step solution

01

Given information.

The curve is $r = \sin 胃 + \cos 胃 for 胃鈭� [0, 蟺]$

02

Arc length

The arc length of the curve $r$ is given by,

localid="1663046444567" $L = {鈭�}_{伪}^{尾} \sqrt{r^{2} + {(\frac{d r}{d 胃})}^{2}} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{{(\sin 胃 + \cos 胃)}^{2} + {(\cos 胃 - \sin 胃)}^{2}} d 胃 [鈭� \frac{d r}{d 胃} = \cos 胃 - \sin 胃] L = {鈭�}_{0}^{2 蟺} \sqrt{(\sin^{2} 胃 + \cos^{2} 胃 + 2 \sin 胃 \cos 胃) + (\sin^{2} 胃 + \cos^{2} 胃 - 2 \sin 胃 \cos 胃)} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{(1 + 2 \sin 胃 \cos 胃) + (1 - 2 \sin 胃 \cos 胃)} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{2} d 胃 L = \sqrt{2} {鈭�}_{0}^{2 蟺} d 胃 L = \sqrt{2} 脳 2 蟺 L = 2 \sqrt{2} 蟺 units$

Thus, the arc length is $L = 2 \sqrt{2} 蟺 units$

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Most popular questions from this chapter

In Exercises 48鈥�55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y = - 3$

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

$r = \cos 4 胃$

Explain why there are infinitely many different ellipses with the same foci.

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

$r^{2} = \cos 胃$

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?

See all solutions

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