Consider the tetrahedron with vertices
(0,0,0)
(0,1,2)
(0,2,0)
(3,0,0)
Find its volume using integration. (No double or triple integrals though)
First, observe that the dimensions of the triangle decrease linearly as you move from z = 0 to z = 2. Therefore,
The length in x is equal to $\displaystyle 3 - 1.5z$
The length in y is equal to $\displaystyle 2 - z$
The area of the triangular cross-section is $\displaystyle 0.5(3 - 1.5z)(2 - z)$.