Volume of Generalized Tetrahedron in R n .I'm having difficulty finding the volume of a...
Volume of Generalized Tetrahedron in .
I'm having difficulty finding the volume of a tetrahedron in .
Find the volume of a generalized tetrahedron in bounded by the coordinate hyperplanes and the hyperplane
In two dimensions, we have . In three dimensions, I got something like .
I am off to a good start?
Answer & Explanation
Your approach is fine, but the allowable range in is , so that should be your integrand. It is probably easier to define the n-volume of a k sided simplex as and recognize that .
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.
It is more accurate to write for two dimensions like this:
For third dimension it will be:
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.