How to solve the following double integral?

Question

We do have:

$\int\int_{D} \frac{xy}{y^{2}-x^{2}}dxdy$

Being D the region limited by the curves:

$x^{2}-y^{2} = 1$

$x^{2}-y^{2} = 4$

$\frac{x^{2}}{16}+\frac{y^{2}}{4} = 1$

$\frac{x^{2}}{4}+y^{2} = 1$

My attempt from the hint:

Should I solve for this integral?

$\int\int_{D} \frac{5x^{2}y^{2}}{y^{2}-x^{2}}dxdy$

How could I get the extremes of the integral from the four equations given?

Thanks

Answer 1

Make a sketch of the domain

and for symmetry we can deduce that the integral over the four part of the region delimited by the curves is equal to zero.

To evaluate the integral for the first quadrant let

• $u=x^2-y^2$
• $v=\frac{x^{2}}{4}+y^{2}$

then $$dudv=|J|dxdy=\begin{vmatrix}2x&-2y\\\frac x 2&2y\end{vmatrix}dxdy=5xy>0$$

thus

$$\int\int_{D} \frac{xy}{y^{2}-x^{2}}dxdy=2\int_1^4dv\int_1^4 -\frac1udu=-6\ln 4$$