# Multiple integral calculus over an algebraic set.

I would like to know, if in general, there is a method of calculus of the following multiple integral : $$I = \int_{\{ (u,v,w) \in \mathbb{R}^3 \ \mid \ u^3 + v^2 + w = -4 \text{ and } u^4 + w^2 = 1 \}} {xyz} \ dx dy dz.$$ Thanks a lot.

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The set $\left\{(u,v,w)\in\mathbb{R}^3 ∣ u^3+v^2+w=−4 \wedge u^4+w^2=1\right\}$ has 3-dimensional Lebesgue measure zero, and hence the value of the integral is zero.
$\{ (u,v,w) \in \mathbb{R}^{3} \ \mid \ u^3 + v^2 + w = -4 \ \ \text{et} \ \ u^4 + w^2 = 1 \} = f^{-1} ( \{ 0 \} ) \bigcap g^{-1} ( \{ 0 \} )$ with $f(u,v,w) = u^3 + v^2 + w + 4 \$ and $\ g(u,v,w) = u^4 + w^2 - 1$, then what to do ? Thank you very much. –  Bryan Dec 27 '12 at 17:19