Chapter 9: Q. 16. (page 756)

complete Example 4 by evaluating the integral
$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ$

Short Answer

Expert verified

The required answer is $8$

Step by step solution

Given information

The integral is $\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ$

The objective is to find the value of the integral.

The integral $\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ .$

$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ d θ} = \int_{0}^{2 π} \sqrt{2 (1 + \cos θ)} d θ = \sqrt{2} \int_{0}^{2 π} \sqrt{(1 + \cos θ)} d θ$

Then,

$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ = \sqrt{2} \int_{0}^{2 π} \sqrt{(1 + 2 \cos^{2} \frac{θ}{2} - 1)} d θ [\sin c e \cos θ = 2 \cos^{2} \frac{θ}{2} - 1] = \sqrt{2} \int_{0}^{2 π} \sqrt{2 \cos^{2} \frac{θ}{2}} d θ = \sqrt{2} \int_{0}^{2 π} \sqrt{2} \cos \frac{θ}{2} d θ$

Find the value of integral

In this case, the supplied function has a negative value in the interval $π$ to $2 π$

Calculate the integral in the range of $0$ to $π$ and multiply it by $2$

$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ = 2 [2 {(2 \sin \frac{θ}{2})}_{0}^{π}] = 8 (\sin \frac{π}{2} - 0) \int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ = 8$

Hence, the value of the integral is $8$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

complete Example 4 by evaluating the integral
$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ$

Short Answer

Step by step solution

Given information

The objective is to find the value of the integral.

Find the value of integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Discrete Mathematics

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

complete Example 4 by evaluating the integral∫02π2+2cosθdθ

Short Answer

Step by step solution

Given information

The objective is to find the value of the integral.

Find the value of integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Discrete Mathematics

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

complete Example 4 by evaluating the integral
$\int_{0}^{2 π} \sqrt{2 + 2 \cos θ} d θ$