Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$.
I set $$u = y-x$$ $$v = z-y$$ $$w = z$$ Isolating for $x$, $y$ and $z$, you get $$x = -u - v+w$$$$y=w-v$$ $$z=w$$
I calculated the Jacobian to be $1$.
The integral would then be:
$$1\dot{}\int_0^3 \int_0^1 \int_0^2 (-u-v+w)(w-v)\,\mathrm{d}u \,\mathrm{d}v\,\mathrm{d}w$$
Which evaluates to $17$, which is incorrect. It's supposed to be $5$. Does anyone know why?
