1. ## jacobian help

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?

2. Your Jacobian does not seem right. I'm getting $\displaystyle |J(u,v)|=\frac{1}{2u}$

3. You can determine either $\displaystyle J(u,v)$ or $\displaystyle J(x,y) \implies J(u,v)=\frac{1}{J(x,y)}$.

Hint: $\displaystyle J(x,y)$ is easier to determine.

4. Originally Posted by Mondreus
You can determine either $\displaystyle J(u,v)$ or $\displaystyle J(x,y) \implies J(u,v)=\frac{1}{J(x,y)}$.

Hint: $\displaystyle J(x,y)$ is easier to determine.
Nice! I am hearing this for the first time.. If you don't mind, can you please elaborate more on this? I didn't clearly get how you determined it.

5. Originally Posted by Sonia
Nice! I am hearing this for the first time.. If you don't mind, can you please elaborate more on this? I didn't clearly get how you determined it.
Oh! now I got it.. the only mistake I was making was that I was integrating u and v values instead of differentiating them.