How do I perform a change of variables?

Question

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x=4$, $z-y=0$, $z-y=1$, $z=0$, $z=5$.

A similar question was asked here. It was never answered, because the OP demonstrated that he/she understood how to perform the change of variables. In that question, the OP sets $u$, $v$, and $w$ equal to expressions involving $x$, $y$, and $z$ and uses those equations to solve the problem. I do not understand how to come up with equations for $u$, $v$, and $w$. Can someone please walk me through that process?

Answer 1

The idea is to choose coordinates such that the limits of integration are somehow 'simpler'. Note that the $0 \leq z \leq 5$ limits are already 'nice' - the issues are with $0 \leq y - x \leq 4$ and $0 \leq z - y \leq 1$.

We would like to have something like $0 \leq u \leq 4$ and $0 \leq v \leq 1$ where $u$ and $v$ are some new coordinates - so do precisely that, and choose $u = y - x$, $v = z - y$, and $w = z$.

With respect to these new coordinates, the domain $D$ becomes $D' = \lbrace (u, v, w) : u \in [0, 4], \,v \in [0, 1], \, w \in [0, 5] \rbrace$. Then we solve this for $(x, y, z)$ by observing that $u + v = z - x$ so $x = w - u - v$, which leads to $(x, y, z) = (w-u-v, w-v, w)$.

Answer 2

You can do the choice $u = y-x,v=z-y$ and $w=z$. Then the bounds in the integral for These new variables are easy, e.g. integral over $u$ goes from 0 to 4. Solve above equation System by $x,y$ and you will get $x(u,v,w),y(u,v,w)$, i.e. the "old coordinates" in dependence of the "new coordinates".

The next step is calculating the Jacobi Matrix $J$ (by computing derivatives of old coordinates by new coordinates) and then its determinant. Because $w=z$ this calculation simplifies a lot.

After this you can compute

$\int \int \int_D xy dV = \int_0^5 \int_0^1 \int_0^4 x(u,v,w)y(u,v,w) det J du dv dw$