Use the given transformation to evaluate the integral. 8xy da r , where r is the region in the first quadrant bounded by the lines y = 2 3 x and y = 3 2 x and the hyperbolas xy = 2 3 and xy = 3 2 ; x = u/v, y = v

Assuming is the region bounded by the curves,



then substituting and changes the boundaries of in the - plane to



so that the region is given by all the points in the set .

Now the area element of the integral over in the - plane is given by the determinant of the following Jacobian matrix:





So the integral is equivalent to