Problem of the Week #75 - September 2nd, 2013

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Chris L T521

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Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem
: If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are constant vectors, $\mathbf{r}$ is the position vector $\langle x,y,z\rangle$ and $E$ is given by the inequalities $0\leq \mathbf{a}\cdot\mathbf{r} \leq \alpha$, $0\leq \mathbf{b}\cdot\mathbf{r} \leq \beta$, $0\leq \mathbf{c}\cdot\mathbf{r} \leq \gamma$, show that
$\iiint\limits_E (\mathbf{a}\cdot\mathbf{r}) (\mathbf{b}\cdot\mathbf{r}) (\mathbf{c}\cdot\mathbf{r}) \,dV = \frac{(\alpha \beta \gamma)^2}{8|\mathbf{a}\cdot(\mathbf{b} \times\mathbf{c})|}$

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