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Chapter 5: Double Integration
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Section 5.4: Changing the Order of Iteration
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Essentials


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The only safe and sure way to convert the iterated integral ${\int}_{a}^{b}{\int}_{y\={y}_{L}\left(x\right)}^{y\={y}_{T}\left(x\right)}f\left(x\,y\right)\mathrm{dy}\mathrm{dx}$ to an equivalent iterated integral in the opposite order, that is, to ${\int}_{c}^{d}{\int}_{x\={x}_{L}\left(y\right)}^{x\={x}_{R}\left(y\right)}f\left(x\,y\right)\mathrm{dx}\mathrm{dy}$, is to infer the region of integration from the first form of the integral, draw that region, and from that sketch, build the second form of the integral with the order of integration reversed. There is no automatic way to accomplish this reversal.

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There are two Maple tools that are useful in deducing the region of integration from a given iterated double integral. The first is a task template for visualizing regions of integration; the second, the inequal command in the plots package. Both tools are demonstrated in the examples below.

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Sometimes reversing the order of integration makes it possible to evaluate an iterated integral that otherwise could not be evaluated in closed form. Sometimes reversing the order of integration will mean that the sum of two or more iterated integrals is the the equivalent of one iteration taken in the opposite order.

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Examples


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Example 5.4.1

Evaluate ${\int}_{0}^{3}{\int}_{0}^{2x}\left({x}^{2}plus;{y}^{2}\right)\mathit{DifferentialD;}y\mathit{DifferentialD;}x$, reverse the order of integration, and evaluate again.

Example 5.4.2

Evaluate ${\int}_{0}^{2}{\int}_{3x}^{6}xy\mathit{DifferentialD;}y\mathit{DifferentialD;}x$, reverse the order of integration, and evaluate again.

Example 5.4.3

Evaluate ${\int}_{0}^{3}{\int}_{{y}^{2}}^{1\+8ysol;3}xy\mathit{DifferentialD;}x\mathit{DifferentialD;}y$, reverse the order of integration, and evaluate again.

Example 5.4.4

Evaluate ${\int}_{1}^{2}{\int}_{{x}^{2}}^{x\+2}\left(x\+2y\right)\mathit{DifferentialD;}y\mathit{DifferentialD;}x$, reverse the order of integration, and evaluate again.

Example 5.4.5

Evaluate ${\int}_{0}^{1}{\int}_{{\sqrt{y}}_{}}^{2}\left(2xplus;3y\right)\mathit{DifferentialD;}x\mathit{DifferentialD;}y$, reverse the order of integration, and evaluate again.



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