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!


1 Answer 1


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 $g=\sqrt{u^2+4v^2}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.