   Chapter 15.1, Problem 32E

Chapter
Section
Textbook Problem

Calculate the double integral.32. ∬ R x 1 + x y d A ,   R = [ 0 , 1 ] × [ 0 , 1 ]

To determine

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

Explanation

Given

The rectangular region is, R=[0,1]×[0,1] .

Calculation:

The value of the given double integral is computed below.

Rx1+xydA=01[01x1+xydy]dx=01[ln(1+xy)]01dx=01[ln(1+x)ln1]dx=01ln(1+x)dx

Integrate the function by using integration by parts.

Let u=ln(1+x) and dv=dx .

Then, v=x .

Rx1+xydA=[xln(1+x)]0101x.11+xdx=[xln(1+x)]01011+x11+xdx=[xln(1+x)]01

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