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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Calculate the double integral.

34. R 1 1 + x + y d A , R = [ 1 , 3 ] × [ 1 , 2 ]

To determine

To calculate: The value of given double integral over the region R.

Explanation

Given

The rectangular region is, R=[1,3]×[1,2] .

Calculation:

Compute the value of double integration as follows.

R11+x+ydA=13[1211+x+ydy]dx=13[ln(1+x+y)]12dx=13[ln(x+3)ln(x+2)]dx=13ln(x+3)dx13ln(x+2)dx

Apply the technique of integration by parts,

Let u1=ln(3+x) and dv1=dx .

Then, v1=x .

Let u2=ln(2+x) and dv2=dy .

Then, v2=y .

The value of the given double integral becomes,

Rx1+x+ydA=[xln(3+x)]1313x.13+xdx[xln(2+x)]13+13x.12+xdx=[xln(3+x)]13133+x33+xdx[xln(2+x)]13+132+x22+xdx=[xln(3+x)]1313dx+31313+xdx[xln(2+x)]13+13dx21312+x

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Chapter 15 Solutions

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