# Change of Variables Problem. How Does One Know What to Select as $u$ and $v$?

I just learned change of variables for double and triple integrals, but I'm having difficulty applying it. When I encounter a problem, I am not sure which variables to set as $u$ and $v$. Unlike single variable substitution ($u$-substitution), there doesn't seem to be any indication of which variables should be selected as $u$ and $v$. In fact, most of the example problems that I have encountered online actually give you the variables $u$ and $v$ to use, rather than having to find them yourself. However, for my studies, I am required to find them myself; this is something I am having trouble with.

Take the following problem I am given:

Evaluate $\iint_D \dfrac{y^4}{x} \ dxdy$ over the region D contained between the parabolas $x = 1 - y^2$ and $x = 4(1 - y^2)$.

The textbook solution says to make $x = v(1 - u^2)$ and $y = u$, but it does not give any indication of how it chose these $u$ and $v$! How did the solution to the above problem find/select $u$ and $v$? As a student, when encountering a problem as above, how can I deduce $u$ and $v$? Is there a proper methodology that is analogous to single variable substitution ($u$ substitution)?

I would greatly appreciate it if people could please take the time to help me understand this methodology.

• Sure. You see, the integration domain -- the "band" between the parabolas $x=1-y^2$ and $x=4(1-y^2)$ in the $x-y$ coordinates is mapped onto a nice rectangular "band", between $v=1$ and $v=4$ in the $u-v$ coordinates. (Check this, by wrtiting $x$ and $1-y^2$ in terms of $u,v$!) This is the motivation behind the choice of these coordinates. – uniquesolution Mar 27 '17 at 14:15