After sketching the region
it should be clear that you can evaluate the required area as
.
Nice graph! Transgalactic, the point of the u, v substitution is that the curves , and , in the xy-plane, become, in the uv-plane, u= 2, u= 4, and u= 7. Of course, those don't form a bounded figure. That's because, as x and y go to 0, u and v go to infinity. So your integral will be for u= 2 to 4, v= 7 to infinity.
Now, when you substitute for x and y, you also have to substitute for u and v: . And that symbol refers to the determinant, .
In order to find that directly, you would have to solve for x and y in terms of u and v. The hint says you don't have to do that. Find, instead, and take the reciprocal.