How to find the volume of a regular tetrahedron using calculus?

V = ∫ dV = ∫∫∫ dx dy dz

This is what I did

first I was thinking to take a cross-sectional equilateral triangle and integrate it from 0 to h, but I could not figure out how to do that. Instead I integrated each variable as below

I integrated x from 0 to a(h-z)/3h and I doubled that

I integrated y from -2L/3 to (h-z)L/3h

I integrated z from 0 to h

But it failed!

I also tried to divide the object into two parts 2L/3 and L/3. I integrated each part separately. I could find the volume of part 2L/3 correctly, but L/3 was not!

I was seeking to get this result (1/3)(1/2)aLh

of course the volume of a regular tetrahedron is a^3 / (6√2) after replacing L and h in terms of a.