Q. 30 In exercise 26-30 Find a definit... [FREE SOLUTION]

Chapter 8: Q. 30 (page 659)

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral
The cardoid $r = 2 - 2 \sin 2 胃 f o r 0 鈮� 胃鈮� 2 蟺$

Short Answer

Expert verified

The integral can be given as $2 \sqrt{2} {鈭�}_{0}^{2 蟺} \sqrt{1 - \sin 胃} d 胃$ and length of the polar curve can be given as $8 \sqrt{2} u n i t s$

Step by step solution

Given information

We are given an equation of cardioid

$r = 2 - 2 \sin 2 胃 f o r 0 鈮� 胃鈮� 2 蟺$

We find the definite integral and evaluate it

We know that length of polar curve can be given as

${鈭�}_{0}^{2 蟺} \sqrt{(f {(胃))}^{2} + (f' {(胃))}^{2}} d 胃$

We are given

$r = 2 - 2 \sin 胃 r' = - 2 \cos 胃$

Substituting the values we get,

localid="1650460683784" $I = {鈭�}_{0}^{2 蟺} \sqrt{{(2 - 2 \sin 胃)}^{2} + {(2 \cos 胃)}^{2}} d 胃 = {鈭�}_{0}^{2 蟺} \sqrt{4 - 8 \sin 胃 + 4 \sin^{2} 胃 + 4 \cos^{2} 胃} d 胃 = {鈭�}_{0}^{2 蟺} \sqrt{8 - 8 \sin 胃} d 胃 = 2 \sqrt{2} {鈭�}_{0}^{2 蟺} \sqrt{1 - \sin 胃} d 胃 = 2 \sqrt{2} {鈭�}_{0}^{2 蟺} (- \cos \frac{胃}{2} + \sin \frac{胃}{2}) d 胃 = 4 \sqrt{2} {[- \sin \frac{胃}{2} - \cos \frac{胃}{2}]}^{2 蟺}_{0} = 4 \sqrt{2} (2) = 8 \sqrt{2} u n i t s$

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91影视

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral
The cardoid $r = 2 - 2 \sin 2 胃 f o r 0 鈮� 胃鈮� 2 蟺$

Short Answer

Step by step solution

Given information

We find the definite integral and evaluate it

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91影视

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral The cardoidr=2-2sin2胃for0鈮�胃鈮�2蟺

Short Answer

Step by step solution

Given information

We find the definite integral and evaluate it

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Geometry

Pure Maths

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral
The cardoid $r = 2 - 2 \sin 2 胃 f o r 0 鈮� 胃鈮� 2 蟺$