Q. 46 Evaluate the iterated integrals ... [FREE SOLUTION]

Chapter 13: Q. 46 (page 1015)

Evaluate the iterated integrals in Exercises 45鈥�48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.
${鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{x}^{\sqrt{蟺}} \cos y^{2} d y d x$

Short Answer

Expert verified

The value of the given integral is: 0

The function $\cos y^{2}$ does not have a simple antiderivative. Therefore it is easier to integrate by reversing the order of integration.

Step by step solution

Step 1. Given Information

An integral, ${鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{x}^{\sqrt{蟺}} \cos y^{2} dy dx$

Step 2. Evaluating the given iterated integrals by reversing the order of integration.

Given integral,

{鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{x}^{\sqrt{蟺}} \cos y^{2} dy dx

Reversing the order of integration,

${鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{0}^{y} \cos y^{2} d x d y = {鈭�}_{0}^{\sqrt{蟺}} (\cos y^{2}) x_{0}^{y} d y = {鈭�}_{0}^{\sqrt{蟺}} y (\cos y^{2}) d y$

Put, $y^{2} = t, 2 y d y = d t,$

We get, ${鈭�}_{0}^{蟺} \frac{(c o s t)}{2} d t$

$= {(\frac{\sin t}{2})}_{0}^{蟺} = \frac{\sin 蟺 - \sin 0}{2} = \frac{0 - 0}{2} = 0$

The function role="math" localid="1653285981950" $\cos y^{2}$ does not have a simple antiderivative. Therefore it is easier to integrate by reversing the order of integration.

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Evaluate the iterated integrals in Exercises 45鈥�48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.
${鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{x}^{\sqrt{蟺}} \cos y^{2} d y d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Evaluating the given iterated integrals by reversing the order of integration.

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91影视

Evaluate the iterated integrals in Exercises 45鈥�48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.鈭�0蟺鈭�x蟺cosy2dydx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Evaluating the given iterated integrals by reversing the order of integration.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate the iterated integrals in Exercises 45鈥�48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.
${鈭�}_{0}^{\sqrt{蟺}} {鈭�}_{x}^{\sqrt{蟺}} \cos y^{2} d y d x$