I have doubt in one question which asks for to evaluate double integrals and here I need to evaluate the Jacobian matrix first.

\iint e^(-xy/2) dy dx, given that x=sqrt(v/u) and y=sqrt(uv)

Bounded by graphs y=x/4, y=2x, y=1/x and y=4/x.

This is what i did:

I took u=y/x and v=xy with limits u:[0.25,2] and v: [1,4].

When I am evaluating the jacobian matrix, m getting 8uv^2/9 as the answer which m incorporating in the equation \iint e^(-v/2). 8uv^2/9 du dv which limits for u and v.

After all this m not getting the correct result which should be approximately 0.97. Where m I possibly going wrong?