How to Simplify the Triple Integral of z^2 Over a Tetrahedron?

In summary, the person is trying to solve a homework problem that involves solving a polynomial, but is having trouble because of the complexity of the algebra. They say that if the problem does not involve a subset of the variables, integrating it over the variables last can help.
  • #1
Seda
71
0

Homework Statement



Essentially, do the volume integral of z^2 over the tetrahedron with vetices at (0,0,0) (1,0,0) (0,1,0) (0,0,1)

The Attempt at a Solution



There seems to be a ton(!) of brute-force algebra involved. Enough to make me question if I'm doing the problem right.
I set up the triple integral of z^2 in the order dzdydx with the following limits of integration.

z=0 to z= 1-x-y
y=0 to y= 1-x
x=0 to x=1

It didn't take to long for me to end up with trying to integrate a humongous polynomial in the second interval.

Evaluating z^3 / 3 at z = 1-x-y was fun enough.

But now after integrating again, I have to evaluate y/3 -xy-y^2/2+xy^2+x^2*y + y^3/3 + 1/3*x^3*y et cetera at y = 1-x seems to be a nightmare.

Am I tackling this the wrong way?
 
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  • #2
No, I think your approach is correct. Though I'm not sure I agree with your xy polynomial. You just have to work through the mess and be careful. If don't have to use a triple integral you can use a shortcut. If A(z) is the area of a cross section of the tetrahedron at a constant value of z, and you can use geometry to get a formula for that area in terms of z, then the triple integral is the integral of A(z)*z^2*dz. Hmm. That sort of suggests that you change the order of integration so you integrate over z last, it might be easier.
 
  • #3
Yeah not only have I found a few mistakes, but i typed some parts in wrong, I was really just writing the polynomial to show how ugly.

Ill try integrating over x last then.
 
  • #4
Seda said:
Yeah not only have I found a few mistakes, but i typed some parts in wrong, I was really just writing the polynomial to show how ugly.

Ill try integrating over x last then.

Z last, I think. You already did X last.
 
  • #5
sorry, i meant z
 
  • #6
Ok i have an answer now, thanks for the help.

Yes, integrating in the different order helped alot. Much cleaner algebra there.
 
  • #7
Right. If your integrand depends on a subset of the variables, integrate over those variables last. It keeps them constants for as long as possible.
 

FAQ: How to Simplify the Triple Integral of z^2 Over a Tetrahedron?

What is a triple integral?

A triple integral is a type of mathematical calculation that involves finding the volume of a three-dimensional shape. It is similar to a regular integral, but instead of finding the area under a curve, it finds the volume under a curved surface.

Why is the triple integral seemingly simple?

The triple integral may seem simple because it follows the same basic principles as a regular integral. However, it becomes more complex when dealing with three-dimensional shapes and multiple variables.

What is causing difficulty with the seemingly simple triple integral?

The difficulty with the seemingly simple triple integral may stem from the complexity of the three-dimensional shape being integrated, the presence of multiple variables, or the need for advanced integration techniques.

What are some strategies for solving a seemingly simple triple integral?

Some strategies for solving a seemingly simple triple integral include breaking it down into smaller, more manageable integrals, using symmetry to simplify the calculation, and applying integration techniques such as substitution or integration by parts.

How can I improve my understanding of triple integrals?

To improve your understanding of triple integrals, it is helpful to practice solving various types of integrals, familiarize yourself with different integration techniques, and seek assistance from a teacher or tutor if needed.

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