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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.8, Problem 14E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Evaluating a Double Integral In Exercises 11-24, evaluate the double integral. See

∫ − 1 2 ∫ − 3 3 ( x 2 − x y 2 )   d y   d x

To determine

To calculate: The double integration ∫−12∫−33(x2−xy2)dydx.

Explanation

Given Information:

The provided integration is ∫−12∫−33(x2−xy2)dydx.

Formula used:

If a binary function F(x,y) is integrable in domain of a≤y≤b and c≤x≤d, the double integration ∫cd∫abF(x,y)dydx can be calculated as follows procedure,

Integrate with respect to y by holding x constant,

∫cd∫abF(x,y)dydx=∫cd[f(x,y)]abdx

Here, function f(x,y) is partial integration of F(x,y) with respect to y variable.

Now, replace the y by limit of integration,

∫cd[f(x,y)]abdx=∫cd[f(x,b)−f(x,a)]dx

Integrate with respect to x,

∫cd[f(x,b)−f(x,a)]dx=[h(x,b)−h(x,a)]cd

Here, the function h(x,b) and is partial integration of f(x,b) with respect to x variable, the function h(x,a) is partial integration of f(x,a) with respect to x variable.

Now, replace the x by limit of integration,

[h(x,b)−h(x,a)]cd=[h(d,b)−h(c,b)−h(d,a)+h(c,a)]

The double integration ∫cd∫abF(x,y)dydx is [h(d,b)−h(c,b)−h(d,a)+h(c,a)]

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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True or False: converges conditionally.

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