Multivariable Substitution.

Question

I am having issues in finding the right substitutions for multi-variable integrals. For instance, evaluate the integral: $\int\int \frac{2x^2+y^2}{xy} dA$ over the region bounded by $y=2\sqrt x$, $y=\sqrt x$, $x^2+y^2=1$ and $x^2+y^2=4$. I have tried the following substitutions $u= x^2+y^2$, $v=\frac{y}{\sqrt x}$. However, when calculating the Jacobian and changing the variables in the original integral in terms of $u$ and $v$, it makes the integral so much more complicated. Can anyone provide a clue as to how to find the correct substitutions as well as any heuristics that might be handy when coming across substitution problems in general.

Answer 1

You're close. You usually want to make substitutions to simplify the region of integration, but often the function you're integrating will provide some clues to get simplification to occur when you introduce the Jacobian determinant.

In this case, you should take $u=x^2+y^2$ and $v=y^2/x$. Then $\dfrac{\partial(u,v)}{\partial(x,y)} = \dfrac{2y}{x^2}\cdot (2x^2+y^2)$. Aha! Of course, when we convert the $xy$-integral into a $uv$-integral, the latter has the "fudge factor" of the inverse Jacobian $\dfrac{\partial(x,y)}{\partial(u,v)} = \dfrac{x^2}{2y(2x^2+y^2)}$. This is going to work out very nicely. Of course, we need an integral in terms of just $u$ and $v$; don't panic and try to express things in terms of $u,v$ until you get to this this point.

Can you take it from here?