Chapter 13: Q. 11 (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.
$\int_{0}^{4} \int_{\sqrt{x}}^{2} y \cos x d y d x$

Short Answer

Expert verified

$\int_{0}^{4} \int_{\sqrt{x}}^{2} y \cos x d y d x = \frac{1 - \cos 4}{2}$

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.

$\int_{0}^{4} \int_{\sqrt{x}}^{2} y \cos x d y d x \Rightarrow \int_{0}^{y^{2}} \int_{0}^{2} y \cos x d y d x$

Evaluate the integral

$I = \int_{0}^{y^{2}} \int_{0}^{2} y \cos x d y d x I = \int_{0}^{2} y [\int_{0}^{y^{2}} \cos x d x] d y I = \int_{0}^{2} y {[\sin x]}_{0}^{y^{2}} d y I = \int_{0}^{2} y \sin y^{2} d y I = \frac{1}{2} {[- \cos y^{2}]}_{0}^{2} I = \frac{1}{2} [- \cos 4 + 1] I = \frac{1 - \cos 4}{2}$

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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.
$\int_{0}^{4} \int_{\sqrt{x}}^{2} y \cos x d y d x$

Short Answer

Step by step solution

Draw the region.

Reversing the order of integration.

Evaluate the integral

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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.∫04∫x2ycosxdydx

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

Pure Maths

Geometry

Calculus

Logic and Functions

Decision Maths

Probability and Statistics

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.
$\int_{0}^{4} \int_{\sqrt{x}}^{2} y \cos x d y d x$