   Chapter 15, Problem 55RE

Chapter
Section
Textbook Problem

Use the transformation u = x − y, v = x + y to evaluate ∬ R x − y x + y d A where R is the square with vertices (0, 2), (1, 1), (2, 2), and (1, 3).

To determine

To evaluate: The integral Rxyx+ydA.

Explanation

Given:

The region R is the square with vertices (0,2),(1,1),(2,2) and (1,3).

The transformation is u=xy, v=x+y.

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)

Calculation:

First find the value of x and y. Add the two transformations will yield, 2x=u+v. Thus, x=12(u+v). Similarly, subtract two transformations will give 2y=uv. Thus, y=12(vu).

Obtain the Jacobian, (x,y)(u,v)=|xuxvyuyv|

Find the partial derivative of x and y with respect to u and v. x=12(u+v) then xu=12 and xv=12 and y=12(vu) then yu=12 and yv=12.

(x,y)(u,v)=|12121212|=12(12)(12)(12)=14+14=12

From the given conditions, rewrite the given integral as given below.

xyx+y=uv

Find the boundary by using the given transformation.

For the point (0,2), u=2,v=2

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