2dfc6e…: “Regardless of the free mountain”

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Let $S=\{(x,y,z):0\le x\le 1, 0\le y\le 1, 0\le z\le 1\}$ and let $F : \R ^3 \to \R$ be the function defined by the rule $F(x,y,z) = x y^2 z^3.$ and let $G : \R ^3 \to \R$ be the function defined by the rule $G(x,y,z) = x y z.$ Which integral is larger?

$\iiint _S F \d V$ $\iiint _S G \d V$

Note that, when $(x,y,z)$ is in $S$ , we have $F(x,y,z) \leq G(x,y,z)$ .