Chapter 13: Q. 12 (page 1082)

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{\sqrt{π}} \int_{x}^{\sqrt{π}} \sin y^{2} d y d x$

Short Answer

Expert verified

$\int_{0}^{\sqrt{π}} \int_{x}^{\sqrt{π}} \sin y^{2} d y d x = 1$

Step by step solution

Draw the region

The region determined by the limits of the given iterated integral is shown below,

Reversing the order of integration

From the above diagram, reversing the order of integration.

$\int_{0}^{\sqrt{π}} \int_{x}^{\sqrt{π}} \sin y^{2} d y d x \Rightarrow \int_{0}^{\sqrt{π}} \int_{0}^{y} \sin y^{2} d y d x$

Evaluate the integral

$I = \int_{0}^{\sqrt{π}} \int_{0}^{y} \sin y^{2} d y d x I = \int_{0}^{\sqrt{π}} \sin y^{2} d y \int_{0}^{y} d x I = \int_{0}^{\sqrt{π}} y \sin y^{2} d y I = \frac{1}{2} {[- \cos y^{2}]}_{0}^{\sqrt{π}} I = \frac{1}{2} [1 + 1] I = 1$

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.

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{\sqrt{π}} \int_{x}^{\sqrt{π}} \sin y^{2} d y d x$

Short Answer

Step by step solution

Draw the region

Reversing the order of integration

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.∫0π∫xπsiny2dydx

Short Answer

Step by step solution

Draw the region

Reversing the order of integration

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{\sqrt{π}} \int_{x}^{\sqrt{π}} \sin y^{2} d y d x$