( 887, #15) Evaluate the integral.

where R is the region in the first quadrant bounded by the lines y = x and y = 3x and the hyperbolas xy = 1, xy = 3;

Solution: This is the region R.

Instead of using the transformation given in the text use the more natural one

u = xy (so u = 1 ..3) and v = y/x (so v = 1..3). Then, solving for x and y in terms of u and v:

The Jacobian is:

Consequently

by: nl