Volume of Generalized Tetrahedron in R n .I'm having difficulty finding the volume of a...

Nathalie Fields

Nathalie Fields

Answered

2022-07-16

Volume of Generalized Tetrahedron in R n .
I'm having difficulty finding the volume of a tetrahedron in R n .
Find the volume of a generalized tetrahedron in R n bounded by the coordinate hyperplanes and the hyperplane x 1 + x 2 + . . . + x n = 1
In two dimensions, we have 0 1 1 x 1 d x 1 . In three dimensions, I got something like 0 1 0 1 x 1 1 x 2 d x 2 d x 1 .
I am off to a good start?

Answer & Explanation

losnonamern

losnonamern

Expert

2022-07-17Added 12 answers

Step 1
Your approach is fine, but the allowable range in x 3 is 1 x 1 x 2 , so that should be your integrand. It is probably easier to define the n-volume of a k sided simplex as V n ( k ) and recognize that V n ( k ) = 0 k V n 1 ( x ) d x.
Step 2
Now each integral is a single one. If you do the first few, you will see a pattern emerge, which you can prove by induction.
Paxton Hoffman

Paxton Hoffman

Expert

2022-07-18Added 6 answers

Step 1
It is more accurate to write for two dimensions like this:
0 1 d x 1 0 x 1 d x 2
Step 2
For third dimension it will be:
0 1 d x 1 0 x 1 d x 2 0 x 2 d x 3
So you can easily write the expression for higher dimensions. My advise is to write the domain bound by hyperplanes more carefully. Your way is too difficult for further integration I guess.

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