Chapter 11: Q12.6P (page 559)

$\int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 - s i n^{2} θ}} d θ .$

Short Answer

Expert verified

The value of integral in elliptic form is $\frac{1}{3} F (\frac{π}{3}, \frac{1}{3}) \approx 0.35499$ .

Step by step solution

Given Information

The value of integration is $\int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 - \sin^{2} θ}} d θ$ .

Definition of elliptic form.

The elliptic form of the integral is defined as $F (\frac{π}{2}, k) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - k^{2} s i n^{2} θ}} d θ$ .

Find the value of Integral.

The value of integration is $\int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 - \sin^{2} θ}} d θ$ .Factor out 9 the equation becomes as follows.

$l = \int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 (1 - \frac{\sin^{2} θ}{9})}} d θ l = \int_{0}^{\frac{π}{3}} \frac{1}{3 \sqrt{(1 - \frac{\sin^{2} θ}{9})}} d θ$

The formula for the beta function is $F (\frac{π}{2}, k) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - k^{2} \sin^{2} θ}} d θ$ .

Equate the above equation with the value of I, the value of I becomes follows.

$l = \frac{1}{3} \int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{(1 - \frac{\sin^{2} θ}{9})}} d θ l = \frac{1}{3} F (\frac{π}{3}, \frac{1}{3}) l \approx 0.35499$

Hence, The value of integral in elliptic form is $\frac{1}{3} F (\frac{π}{3}, \frac{1}{3}) \approx 0.35499$ .

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 Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$\int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 - s i n^{2} θ}} d θ .$

Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Find the value of Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Kinematics Physics

Force

Physics of Motion

Particle Model of Matter

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

∫0π319-sin2θdθ.

Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Find the value of Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Kinematics Physics

Force

Physics of Motion

Particle Model of Matter

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

$\int_{0}^{\frac{π}{3}} \frac{1}{\sqrt{9 - s i n^{2} θ}} d θ .$