( 887, #21) Evaluate the integral by making an appropriate change of variables.

where R is the trapezoidal region with vertices ( 1,0), ( 2, 0), ( 0, 2) , and ( 0, 1).

Solution: This is the trapezoidal region R.

If we set u = x + y and v = x - y, then:

Working out the image of the trapezoid R:

On the line x + y = 1, x = 0..1

u = x + y = 1
v = x - y = x - ( 1 - x ) = 2x - 1 , v = -1..1

The image of the line x+y = 1 is the line u = 1, v = -1..1:

On the line y = 0, x = 1..2.

u = x
v = x (v = u)

So, the image is the line v = u, where u = 1..2. We add it to the previous one:

Continuing:

On the line x + y = 2, x = -2 ..0.
u = x + y =2
v = x - y = x - ( 2 - x ) = 2x - 2, v = 2 .. -2 x

Finally:

On the line x = 0, y = 1..1.
u = y
v = -y (v = -u)

So, the image of R to is a polygon defined by the lines u = 1, u = 2, v = u and v = -u.

As for the Jacobian,

Thus:

We can now evaluate the integral:

Since the domain of integration is symmetric with respect to the u axis, and since
cos( v/(-u)) = cos(v/u), this last integral is equal to

by: nl