Volume integral of a function over tetrahedron

In summary, the conversation discusses the volume integral of the function f(x,y,z)=xyz^2 over a tetrahedron with specified corners. The solution to the integral is given and it is noted that the small answer may be due to the behavior of the function within the tetrahedron. The physical significance of the volume integral is questioned, but it is concluded that it is primarily a mathematical calculation.
  • #1
physkim
6
1

Homework Statement


Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$
over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$

Homework Equations


I was able to solve it mathematically, but still can't 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?

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 !
 
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  • #2
physkim said:
$$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$
I think one of those points should be the origin.
physkim said:
What is the physical significance for calculating the volume integral of an arbitrary function over a geometrical shape?
It just means you are summing the value of the function at every point within the specified volume.
physkim said:
still can't figure out why the answer is so small.
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.
 
  • #3
blue_leaf77 said:
I think one of those points should be the origin.
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?
 

Related to Volume integral of a function over tetrahedron

1. What is a volume integral?

A volume integral is a mathematical concept used in calculus to calculate the total volume of a three-dimensional space. It involves dividing the space into infinitesimal volume elements and summing them up using integration.

2. What is a tetrahedron?

A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices. It is a type of pyramid with a triangular base and three triangular faces connecting to a single point.

3. How do you calculate the volume integral of a function over a tetrahedron?

To calculate the volume integral of a function over a tetrahedron, you first need to set up the integral using the function's formula and the limits of the tetrahedron's volume. Then, you can solve the integral using integration techniques, such as the substitution method or the integration by parts method.

4. What is the significance of calculating the volume integral of a function over a tetrahedron?

The volume integral of a function over a tetrahedron is significant because it allows us to find the total volume of a three-dimensional space, which can be useful in various fields such as physics, engineering, and mathematics. It also helps us understand the behavior and properties of the function within the given volume.

5. Are there any real-world applications of calculating the volume integral of a function over a tetrahedron?

Yes, there are several real-world applications of calculating the volume integral of a function over a tetrahedron. For example, in fluid mechanics, it is used to determine the total mass of a fluid within a given volume. In physics, it is used to calculate the moment of inertia of a solid object. In engineering, it is used to find the center of mass of a three-dimensional structure, among many other applications.

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