Question

Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through ​, ​, and .

Answer 1

Let's assume a plane

`ax+by+cz=1`

with each of a, b, and c greater than 0. Then the plane has intercepts with coordinates

`y=z=0 \implies ax = 1 \implies x = \frac1a`

`x=z=0 \implies by = 1 \implies y = \frac1b`

`x=y=0 \implies cz = 1 \implies z = \frac1c`

The volume of the tetrahedron is then

`\displaystyle \int_0^(\frac1a) \int_0^{\frac{1-ax}b} \int_0^{\frac{1-ax-by}c} dz \, dy \, dx = \boxed{\frac1{6abc}}`

i.e. 1/6 times the product of the intercepts' non-zero coordinates