Evaluate the following integralR6x(8x - y) In (8x - y)dAwhere R is the region bounded by the graphs of y =the change of variables u = 8x − y, v =6x + y.8x – e6, y = 8x – e³, y = −6x + 3, and y= -6x + 2. Use

Given that, where R is the region bounded by the graphs of Using the change of…

Answered: Evaluate the following integral 6x + SS…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Evaluate the following integral
R
6x
(8x - y) In (8x - y)
dA
where R is the region bounded by the graphs of y =
the change of variables u = 8x − y, v =
6x + y.
8x – e6, y = 8x – e³, y = −6x + 3, and y
= -6x + 2. Use

Expert Solution

Step 1: First, we write down the given information.

Given that

$integral integral subscript R fraction numerator 6 x plus y over denominator left parenthesis 8 x minus y right parenthesis ln left parenthesis 8 x minus y right parenthesis end fraction d A$ , where R is the region bounded by the graphs of $y equals 8 x minus e to the power of 6 comma space y space equals space 8 x minus e cubed comma space y space equals space minus 6 x plus 3 comma space and space space y space equals space minus 6 x plus 2.$

Using the change of variables $u equals 8 x minus y comma space v space equals space 6 x plus y.$

$x equals u over 14 space plus space v over 14 comma space space y space equals space fraction numerator 4 space v over denominator 7 end fraction space minus space fraction numerator 3 space u over denominator 7 end fraction comma J open parentheses x comma y close parentheses space equals space open vertical bar table row cell 1 over 14 end cell cell 1 over 14 end cell row cell 4 over 7 end cell cell negative 3 over 7 end cell end table close vertical bar space equals space minus space 3 over 98 minus 4 over 98 equals negative 1 over 14 space comma 8 x minus y equals e to the power of 6 comma space 8 y minus 6 space equals e cubed comma space y plus 6 x space equals space 3 comma space and space straight y space plus 6 straight x space equals space 2.$