   Chapter 15.9, Problem 24E

Chapter
Section
Textbook Problem

Evaluate the integral by making an appropriate change of variables.24. ∫∫R (x + y)ex2 + y2dA. Where R is the rectangle enclose by the lines x – y = 0, x – y = 2, x + y = 0, and x + y = 3

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

The integral is R(x+y)ex2y2dA where the rectangle enclosed by the lines xy=0, xy=2, x+y=0 and x+y=3.

Property used: Change of Variable

Change of Variable in double integral is given by, Rf(x,y)dA=Sf(x(u,v),y(u,v))|(x,y)(u,v)|dudv (1)

Definition used: Jacobian transformation

(x,y)(u,v)=|xuxvyuyv|=xuyvxvyu

Calculation:

Let u=x+y and v=xy

Obtain the value of |(x,y)(u,v)|.

Find the partial derivative of x  and y with respect to u and v. u=x+y then ux=1 and uy=1 and v=xy then vx=1 and vy=1.

T1=(u,v)(x,y)=|uxuyvxvy| (2)

Substitute the corresponding values in equation (2),

T1=|1111|=1(1)1(1)=11=2

This is in the form of inverse of Jacobian transformation (T1) but obtain Jacobian transformation (T). Therefore the value of Jacobian 2  will be 12

R is the image of the rectangle enclosed by the lines u=0, u=3, v=0 and v=2

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