# Change of Coordinates Multivariable Calculus

Consider the region $D$ of the plane in the first quadrant bounded by the hyperbolas $x^2-y^2=1$, $x^2-y^2=4$ and the circles $x^2+y^2=4$ and $x^2+y^2=9$. Use change of coordinates to find the double integral $$\iint_Dx\;dA$$ I've graphed the region, but I don't know how to find the bounds.

• Welcome to MSE. Please use Latex to write the Math. – nls Nov 5 '16 at 15:55
• Hint: $x=\sqrt{\frac{u+v}{2}}$, $y=\sqrt{\frac{v-u}{2}}$. – Kuifje Nov 5 '16 at 17:04

If you perform the change of variables \begin{cases} x=\sqrt{\frac{v+u}{2}}\\ y=\sqrt{\frac{v-u}{2}}\\ \end{cases} Your region $D$ becomes much more friendly: it is bounded by $u=1$, $u=4$, $v=4$, $v=9$, i.e. it is a rectangle in the $(u,v)$-plane.
The Jacobian of such a transformation is $\frac{1}{2(u+v)}$.
It follows that $$\iint_D x\; dA = \int_1^4\int_4^9\sqrt{\frac{v+u}{2}}\frac{1}{2(u+v)}\;dvdu$$