MHB Vector calculus position vector

[Chris L T521]

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

-----

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
\[\iiint\limits_E (\mathbf{a}\cdot\mathbf{r}) (\mathbf{b}\cdot\mathbf{r}) (\mathbf{c}\cdot\mathbf{r}) \,dV = \frac{(\alpha \beta \gamma)^2}{8|\mathbf{a}\cdot(\mathbf{b} \times\mathbf{c})|}\]

-----

 

[Chris L T521]

No one answered this week's question. Since the solution was such a pain (sorry about not working it out prior to posting the problem), I've typed up this week's solution as a pdf [since I don't think it would be able to fit nicely in one post].

You can find the solution in my public dropbox folder by https://dl.dropboxusercontent.com/u/25818055/MHB_POTW_University_Wk75.pdf.