The limits of integration, in terms of x and y are: y= 1, y= 2, and x= 1/y, x= y.
Since x= u/v and y= uv, those become uv= 1, uv= 2, and u/v= 1/uv, u/v= uv. The first two give v= 1/u and and v= 2/u while the last two reduce to and .
Drawing those lines on a uv- graph, you can see that u= 1 and v= 1 intersect at (1, 1) on the graph of v= 1/u and intersect the graph of y= 2/v at u= 1, v= 2 and u= 2, v= 1. Essentially, then, the region you want to integrate over, in the uv-plane, is bounded by u= 1, v= 1, and v= 2/u. If u ranges from 1 to 2 then, for each u, v ranges from 1 to 2/u.