   Chapter 15, Problem 57RE

Chapter
Section
Textbook Problem

Use the change of variables formula and an appropriate transformation to evaluate ∬ R x y   d A , where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, −1).

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

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

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:

Use the transformation u=xy, v=x+y. 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.

xy=12(u+v)12(uv)=14(u2v2)

Find the boundary by using the given transformation.

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

For the point (1,1), u=2,v=0.

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

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