Volume by integrals of an icosahedron

Question

I want to know if there is an easy way to find the volume of a convex icosahedron using integrals. My problem is that I don't know how to determine what the integration limits are.

Any help will be appreciate.

Answer 1

You probably want to use the divergence theorem. That is, the divergence theorem states that $$\iiint_V \nabla \cdot F dV = \iint_S F \cdot n dS,$$ where $S$ is the surface of the region enclosed by $F$. Choose $F=\frac{1}{3}(x,y,z)$ (or anything such that the divergence is 1) and compute the surface integral over each face of your polyhedron.

Integral over each face will be integrating over a polyhedron in the plane. So in essence, you integrate over the projection of the faces.