Chapter 13: Q 58. (page 1014)

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{0}^{3} \int_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$

Short Answer

Expert verified

The value of integral is $18$ .

Step by step solution

Given Information

Iterated integral is $\int_{0}^{3} \int_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$

Calculation of Graph

The graph is as below:

Simplification

Changing the order of integral

$\int_{0}^{3} \int_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y = \int_{0}^{3} \int_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x$

Therefore

$\int_{0}^{3} \int_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x = \int_{0}^{3} [y]_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d x$

$= \int_{0}^{3} (9 - x^{2}) d x$

$\Rightarrow \int_{0}^{3} \int_{0}^{\sqrt{9 - x^{2}}} \sqrt{9 - x^{2}} d y d x = {[9 x - \frac{x^{3}}{3}]}_{0}^{3}$

$18$

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Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{0}^{3} \int_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$

Short Answer

Step by step solution

Given Information

Calculation of Graph

Simplification

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Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.∫03∫09-y29-x2dxdy

Short Answer

Step by step solution

Given Information

Calculation of Graph

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Probability and Statistics

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
$\int_{0}^{3} \int_{0}^{\sqrt{9 - y^{2}}} \sqrt{9 - x^{2}} d x d y$