Let's do this. A general case you may find helpful. Tetrahedron-wise.

The tetrahedron in the first octant bounded by the coordinate planes and TKH's plane:

Of course,

6 different integrals and they all evaluate to .

I hope.

WHEW!!!

Evaluate. They should all be (abc)/6.

I think your second one is the same only arranged differently in the plane.