   Chapter 7.8, Problem 24E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### 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 ∫ 0 ∞ ∫ 0 ∞ x y e − ( x 2 + y 2 ) d x   d y

To determine

To calculate: The double integration 00xye(x2+y2)dxdy.

Explanation

Given Information:

The provided integration is 00xye(x2+y2)dxdy.

Formula used:

If a binary function F(x,y) is integrable in domain of ayb and cxd, the double integration cdabF(x,y)dydx can be calculated as follows procedure,

Integrate with respect to y by holding x constant,

cdabF(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 cdabF(x,y)dydx is [h(d,b)h(c,b)h(d,a)+h(c,a)].

Calculation:

Consider the double integration,

00xye(x2+y2)dxdy

Convert, unbounded limit into bounded limit by introducing parameter a and b,

limab0a0bxye(x2+y2)dxdy

Integrate with respect to x by holding y constant,

limab0a

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