Question: Consider the one-to-one transformation $(u; v;w) \to (x; y; z)$ defined by the equations $u = x + y + z; uv = y + z; uvw = z;$ which maps the unit cube $U$ defined by $0 \lt u \lt 1, 0 < v < 1, 0 < w < 1$ onto the tetrahedron $T$ defined by $x > 0, y > 0, z > 0, x + y + z < 1.$
I need to evaluate the integral $\int \int \int e^{-(x+y+z)^3} \;dz \;dy \;dz$ changing the variables.
For the Jacobian I got $u^2v(1-2v),$ then the integral would be
$\int_0^1 \int_0^1 \int_0^1 e^{-u^3} |u^2 v (1-2v) | \; du \; dv \; dw$
Am I correct so far?
I'm struggling to integrate $e^{-u^3}$ from here.