Chapter 13: Q. 13 (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.
$\int_{0}^{9} \int_{\sqrt{y}}^{3} \frac{1}{1 + x^{3}} d x d y$

Short Answer

Expert verified

$\int_{0}^{x^{2}} \int_{0}^{3} \frac{1}{1 + x^{3}} d x d y = \frac{1}{3} \ln 28$

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.

$\int_{0}^{9} \int_{\sqrt{y}}^{3} \frac{1}{1 + x^{3}} d x d y \Rightarrow \int_{0}^{x^{2}} \int_{0}^{3} \frac{1}{1 + x^{3}} d x d y$

Evaluate the integral

$I = \int_{0}^{x^{2}} \int_{0}^{3} \frac{1}{1 + x^{3}} d x d y I = \int_{0}^{x^{2}} d y \int_{0}^{3} \frac{1}{1 + x^{3}} d x I = \int_{0}^{3} \frac{x^{2}}{1 + x^{3}} d x$

Substitute,

$1 + x^{3} = t 3 x^{2} d x = d t d x = \frac{1}{3 x^{2}} d t When x = 0, t = 1 + 0 = 1 When x = 3, t = 1 + 3^{3} = 28$

Therefore,

$I = \frac{1}{3} \int_{1}^{28} \frac{1}{t} d t I = \frac{1}{3} {[\ln t]}_{1}^{28} I = \frac{1}{3} \ln 28$

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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.
$\int_{0}^{9} \int_{\sqrt{y}}^{3} \frac{1}{1 + x^{3}} d x d y$

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.∫09∫y311+x3dxdy

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

Decision Maths

Statistics

Geometry

Discrete Mathematics

Mechanics Maths

Logic and Functions

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.
$\int_{0}^{9} \int_{\sqrt{y}}^{3} \frac{1}{1 + x^{3}} d x d y$