What is the volume of the 3D-object formed by the intersection between the hyperplane $x+y+z+w=2$ and a 4D cube with verticies that have coordinates 0 or 1?
By computing all the pairwise distances between the points on the plane and the cube (there are 6 points) we can quickly find that the desired 3D object is an octahedron, with a volume of $\frac 43.$
I was wondering what would the multivariable calculus method be to determine this volume? I was thinking of defining $w = 2 - x - y - z$ and evaluating something such as $$2\iiint_{0\leq x,y,z\leq1 \,\,\, \text{and} \,\,\, 1 \leq x+y+z\leq 2} dA$$ but I was unsure about how to compute the actual bounds to this integral or if this integral is even correct. Is there a better way on how to set this integral up, or is there a trick to computing the above integral?