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 = 1In 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?

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Volume of Generalized Tetrahedron in       R    n  .I&#039;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    =  1In 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?

losnonamern · Accepted Answer

Step 1Your 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 2Now 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 · Accepted Answer

Step 1It is more accurate to write for two dimensions like this:      ∫    0    1    d      x    1        ∫    0                  x        1              d      x    2  Step 2For 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.

Find the volume of a generalized tetrahedron in mathbbR^n bounded by the coordinate hyperplanes and the hyperplane x_1+x_2+cdots+x_n=1.

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