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Homework Help: Volume integral of a function over tetrahedron

  1. Jan 19, 2016 #1
    1. The problem statement, all variables and given/known data
    Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$
    over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$
    2. Relevant equations
    I was able to solve it mathematically, but still cant figure out why the answer is so small.
    I only understand that if f(x,y,z) is the density, then the triple integral is the mass.
    What is the physical significance for calculating the volume integral of an arbitrary function over a geometrical shape?

    3. The attempt at a solution

    $$\int_{0}^{1} \int_{0}^{1-y} \int_{0}^{1-x-y} xyz^2 dz dx dy =\frac{1}{2520}$$

    Big thanks in advance !
  2. jcsd
  3. Jan 19, 2016 #2


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    Homework Helper

    I think one of those points should be the origin.
    It just means you are summing the value of the function at every point within the specified volume.
    For now I assume that your calculation hides no mistake. The minuscule value of the integral might be caused by the behavior of your ##f(x,y,z)## within the specified tetrahedron. Look, the biggest value of either ##x##, ##y##, and ##z## within this tetrahedron is unity, therefore ##f(x,y,z)=xyz^2## cannot be bigger than unity (although I haven't calculated what the maximum value is, but certainly the maximum value cannot be bigger than 1). In fact, the values of coordinates are multiplied in ##f(x,y,z)## which makes this function sufficiently small if you remember that multiplication between two or more numbers gives a number which is smaller than the smallest number being multiplied.
  4. Jan 19, 2016 #3
    Yes, sorry, I made a mistake while typing.

    You are right! the answer is so small because I am multiplying a fraction to a fraction and again to a fraction squared.

    So there is no specific physical significance (I am asking this because this is a problem from my EM Theory assignment), only pure mathematical calculation? I should simply treat this integral as a mathematical problem, rather than trying to combine it with physical quantities?
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