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Integrate f(x, y, z)?

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.

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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
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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.