Volume of a simplex

Question

Find the hypervolume of the hypersolid in 4-space $\mathbb{R}^4$ consisting of the points $(w,x,y,z)$ that satisfy $w\ge0,x\ge0,y\ge0,z\ge0$ and $w+2x+3y+4z\le6$

---

This is a problem from our undergrad math competition a few years ago. And I just couldn't wrap my head around the concept of four dimension. I guess we are using quadruple integral to do this? Any help would be appreciated.

Answer 1

If instead you had the simplex $$ x,y,z,w\ge 0 \quad\text{and}\quad x+y+z+w\le 1, $$ then its volume would be $\frac{1}{4!}=\frac{1}{24}$.

Now, what do the coefficients 2,3 and 4 are doing?

Answer. Divide by 2,3 and 4.

What does 6, in the position of 1 do?

Answer. Multiply by $6^4$.

Altogether

Volume of simplex $=\frac{6^4}{24^2}=\frac{9}{4}.$

Answer 2

It is the quadruple integral: the set up is:

$$V = \int^{6}_{0}\int^{\frac{6-w}{2}}_{0}\int^{\frac{6-w-2x}{3}}_{0}\int^{\frac{6-w-2x-3y}{4}}_{0}1\,dz\,dy\,dx\,dw$$