How Do You Calculate the Volume of a Tetrahedron with Given Vertices?

[evilpostingmong]

Homework Statement

Find the volume of a tetrahedron with vertices (0,0,0) (0,0,1) (0,2,0) (2,2,0)

Homework Equations

The Attempt at a Solution

The tetrahedron is bounded by the functions z=1-x/4-x/4, z=1-y/2, and y=x
I integrated z=1-x/4-y/4 first and z=1-y/2 second and subtracted the result
(they were both negative so I subtracted the result from z=1-x/4-y/4 from z=1-y/2).
The limits of integration I used when integrating 1-x/4-y/4 were y<x<2 and 0<y<2
and for 1-y/2, y<x<2 and 0<y<2.

 

[mighty2000]

The result I got was 1/6.



Hello! Great job on attempting to solve this problem. However, there are a few things to note about your solution. First, the equations you listed are not the equations of the functions that bound the tetrahedron. The correct equations are z=1-x/2-y/2, z=1-y/2, and y=x. Second, when integrating, you need to consider the limits of integration for each variable separately. For example, for the first integral, you would have 0<x<2 and 0<y<x. And for the second integral, you would have 0<y<2 and 0<x<2. Finally, the result you got, 1/6, is not the volume of the tetrahedron. It seems that you may have made a mistake in your integration. I suggest rechecking your work and also using the correct equations for the boundaries of the tetrahedron. Good luck!