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Chapter 12: Problem 64

Explain why it is sometimes an advantage to change the order of integration.

Short Answer

Expert verified
Changing the order of integration can sometimes simplify the computation by avoiding complex integration limits or dealing with functions that are difficult to integrate in their current form.

Step by step solution

01

Understand the Concept of Order of Integration

The order of integration refers to the sequence in which the integrals in a multiple integral are performed. When computing a double or triple integral, you first integrate with respect to one variable, then another, etc. This is the order of integration. The order can be changed depending on the nature of the problem at hand.
02

Consider a Scenario with Complex Integration Limits

Consider the example where you have a double integral over an area that has curvy, complex boundaries when expressed in one order, but easy boundaries when rearranged. Changing the order of integration can then convert this complex limit into simpler terms, allowing for easier computation.
03

Dealing with Unintegrated Functions

In some scenarios, the integrand may be in a form where it's impossible or very difficult to integrate with respect to the variable that you're trying to integrate first according to the initial order of integration. Changing the order might allow the integration to become possible or considerably more straightforward.

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Most popular questions from this chapter

In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{1}^{4} \int_{1}^{-\sqrt{x}} 2 y e^{-x} d y d x $$

Find the mass of the lamina described by the inequalities, given that its density is \(\rho(x, y)=x y .\) (Hint: Some of the integrals are simpler in polar coordinates.) $$ x \geq 0,0 \leq y \leq 9-x^{2} $$

The value of the integral \(I=\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x\) is (a) Use polar coordinates to evaluate the improper integral $$ \begin{aligned} I^{2} &=\left(\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x\right)\left(\int_{-\infty}^{\infty} e^{-y^{2} / 2} d y\right) \\ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(x^{2}+y^{2}\right) / 2} d A \end{aligned} $$ (b) Use the result of part (a) to determine \(I\). For more information on this problem, see the article "Integrating \(e^{-x^{2}}\) Without Polar Coordinates" by William Dunham in Mathematics Teacher. To view this article, go to the website ww.matharticles.com

In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{\cos y} y d x $$

In Exercises \(59-62,\) use a computer algebra system to approximate the iterated integral. $$ \int_{0}^{2} \int_{x}^{2} \sqrt{16-x^{3}-y^{3}} d y d x $$
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