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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.

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    I think your Jacobian is wrong. I got $J(u,v,w)=u^2v$.2013-06-19

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