Q. 27 The arc length of polar function... [FREE SOLUTION]

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Chapter 9: Q. 27 (page 775)

The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = 1 鈭� \sin 胃 for 胃鈭� [0, 2 蟺]$

Short Answer

Expert verified

The length of an arc is $\sqrt{2} units$

Step by step solution

Given information.

The curve is $r = 1 - \sin 胃 for 胃鈭� [0, 2 蟺]$

Arc length.

The length of an arc is given by,

$L = {鈭�}_{伪}^{尾} \sqrt{r^{2} + {(\frac{d r}{d 胃})}^{2}} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{{(1 - \sin 胃)}^{2} + {(- \cos 胃)}^{2}} d 胃 [鈭� \frac{d r}{d 胃} = - \cos 胃] L = {鈭�}_{0}^{2 蟺} \sqrt{(1 + \sin^{2} 胃 - 2 \sin 胃) + (\cos^{2} 胃)} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{(2 - 2 \sin 胃)} d 胃 [\sin^{2} 胃 + \cos^{2} 胃 = 1] L = {鈭�}_{0}^{2 蟺} \sqrt{2 (1 - \sin 胃)} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{2 (1 - \cos (\frac{蟺}{2} - 胃))} d 胃 L = {鈭�}_{0}^{2 蟺} \sqrt{2 (2 \sin^{2} (\frac{蟺}{4} - \frac{胃}{2}))} d 胃 L = 2 {鈭�}_{0}^{2 蟺} \sin (\frac{蟺}{4} - \frac{胃}{2}) d 胃 L = 2 {[\frac{1}{2} \cos (\frac{蟺}{4} - \frac{胃}{2})]}_{0}^{2 蟺} L = [\cos (\frac{蟺}{4} - \frac{2 蟺}{2}) - \cos (\frac{蟺}{4} - \frac{0}{2})] L = [\cos (\frac{- 3 蟺}{4}) - \cos (\frac{蟺}{4})] L = [- \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}] L = |- \frac{2}{\sqrt{2}}| L = \sqrt{2} units$

The arc length isrole="math" localid="1663047738225" $\sqrt{2} units$

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91影视

The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = 1 鈭� \sin 胃 for 胃鈭� [0, 2 蟺]$

Short Answer

Step by step solution

Given information.

Arc length.

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91影视

The arc length of polar functions: Find the arc lengths of the following polar functions. r=1鈭�sin胃for胃鈭�[0,2蟺]

Short Answer

Step by step solution

Given information.

Arc length.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Decision Maths

Theoretical and Mathematical Physics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

The arc length of polar functions: Find the arc lengths of the following polar functions.
$r = 1 鈭� \sin 胃 for 胃鈭� [0, 2 蟺]$