Integrate f(x, y, z)?

Question

Integrate f(x, y, z)=x^2+y^2-z over the tetrahedron with vertices (0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3).

 

Please show all your work.

Answer 1

There are two ways do evaluate this volume.

 

The first is to use the high school formula for the volume of a pyramid.  This is V = (1/3) base_area x height.

The base area (in the x y plane) is 1/2.  The height = 3,   so   V =  (1/3) (1/2) 3  = 1/2.

 

The calculus method is to notice that we have a stack of isosceles right triangles all normal to the z axis. 

The side, s, of these triangles is  s(z) =  (3-z)/3.    Since the area of an isosceles triangle of side s is

    (1/2) s² ,

we must integrate  (1/2) [ (3-z)/3 ]²  dz    from 0 to 3.    This integral is elementary because the integrand is

a polynomial form.    It evaluates to 1/2  as expected.