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I must evaluate $\iint_B(x^2+y^2)dxdy$ using the change of variables $u=x^2-y^2,v=xy$, where $B$ is the region in the first quadrant bounded by $xy=1,xy=3,x^2-y^2=4,x^2-y^2=1$.

I know that we must have: $$\iint_Df(x,y)dxdy=\iint_Sg(u,v) \left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv$$ I have computed this Jacobian to be $\frac{1}{4v}$, and $S$ will be represented by $1 \leq u \leq 4, 1 \leq v \leq 3$. However, I am unsure of how to find $g(u,v)$ in this scenario; from other examples I've seen, it's usually a simple substitution. Any help in how to transform $f(x,y)=x^2+y^2$ into a form $g(u,v)$ would be greatly appreciated. Thank you!

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Let $g=x^2+y^2$. Then

$$\begin{cases} g+u=2x^2 \\ g-u=2y^2 \end{cases} \implies g^2-u^2=4v^2 $$

Since $g$ is always nonnegative, we conclude $g=\sqrt{u^2+4v^2}$.

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