Evaluate double integral bounded by lines and hyperbolas

Question

Evaluate the integral $$\iint_R x^2 y^2 dx dy,$$ where $R$ is the bounded portion of the first quadrant bounded by the lines $y=x, y=4x$ and the hyperbolas $xy=1$ and $xy=2.$

Based on the graph of this region, I could split the double integral into three parts based on the range of $x$ values ($1/2$ to $\sqrt 2/2,$ $\sqrt 2/2$ to $1$, and $1$ to $\sqrt 2$). This is feasible, but I'm wondering if there's a simpler way?

Answer 1

Hint:

You are completely right. There is a simpler way, the method of variable transformation. Let: $$x=\dfrac uv, y=uv,\tag1 $$ so that $$u^2=xy, v^2=\frac yx.$$

In the variables $u,v $ the integration will be over a rectangular region. It remains only to compute the transformation Jacobian.

Can you take it from here?

The Jacobian for the transformation $(1) $ reads:$$J=\left|\begin {matrix}\dfrac1v&-\dfrac u {v^2}\\v&u\end {matrix}\right|=\dfrac {2u}v,$$so that one finally obtains:$$\iint_R x^2 y^2 dx dy=2\int_1^2\frac {dv}v\int_1^\sqrt2u^5du.$$