Chapter 19: Chapter 19

Problem 565

Express $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}$ integrals where $0 \leq \mathrm{x} \leq(\pi / 6)$.

Short Answer

Expert verified

The given integral can be expressed in terms of an incomplete elliptic integral of the second kind as \[\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).\]

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Recall the definition of an incomplete elliptic integral of the second kind

An incomplete elliptic integral of the second kind, denoted as $E(\phi, k)$, is defined as: \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta}.\]

Step 2: Compare the given integral to the elliptic integral definition

Given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Comparing it with the definition of an elliptic integral of the second kind: \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},\] We can notice the similarities as both the integrals have the same integral form. Now, in order to rewrite the given integral in terms of elliptic integrals, we need to identify $\phi$ and $k$.

Step 3: Identify $\phi$ and $k$ from the given integral

The given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Comparing it with the definition of an elliptic integral of the second kind \[E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},\] we can make the following identifications: 1. $\phi = \mathrm{u}$ 2. $k^2 = 4$ With these identifications, we have $k = 2$.

Step 4: Rewrite the given integral in terms of elliptic integrals

Now that we have identified $\phi$ and $k$, we can rewrite the given integral in terms of the incomplete elliptic integral of the second kind. The given integral is: \[\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.\] Using our identifications from Step 3, we can express the integral as: \[\mathrm{x} E(\mathrm{u}, 2).\] Since $0 \leq \mathrm{x} \leq (\pi / 6)$, we can conclude: \[0 \leq \mathrm{u} \leq (\pi / 6).\] So, the expression of the given integral in terms of an incomplete elliptic integral of the second kind is: \[\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).\]

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Select your language

Chapter 19: Chapter 19

Express $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}\( integrals where \)0 \leq \mathrm{x} \leq(\pi / 6)$.

Short Answer

Step by step solution

Unlock all solutions

Step 1: Recall the definition of an incomplete elliptic integral of the second kind

Step 2: Compare the given integral to the elliptic integral definition

Step 3: Identify \(\phi\) and \(k\) from the given integral

Step 4: Rewrite the given integral in terms of elliptic integrals

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Advanced Calculus’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Geometry

Probability and Statistics

Calculus

Statistics