What will be the equation of the tetrahedron that passes through the points (0,0,0),(1,0,0),(0,1,0) and (0,0,1) ?
help will be appreciated.
Well basically i have to evaluate a triple integral xyz over a region D which is a tetrahedron with the vertices (0,0,0),(1,0,0),(0,1,0) and (0,0,1). So basically i need the equation of the plane that will pass through these points. Maybe i wrote it in the wrong way. Will it be x+y+z=1 or x-y+z=1 ?
My professor used the latter one to solve the integral i can't figure out why?
I was just working with tetrahedrons and similar objects. You can actually define a tetrahedron as the subset of $\displaystyle \mathbb{R}^4$ with $\displaystyle x,y,z,w\ge{0}$ and $\displaystyle x+y+z+w=1$, a definition that generalizes easily to any number of dimensions.
But looking at your problem, for the point (0,1,0), x-y+z is -1, so that doesn't work. Based on what you said, x+y+z=1 is the correct equation. If you can post the exact problem and the professor's solution, maybe we can figure out what happened.
Since you're setting up a triple integral, you will want limits of integration, which will look like:
$\displaystyle \int_0^1{\int_0^{1-z}{\int_0^{1-y-z} xyz\ dx\ dy\ dz}}$
depending on what order you choose to integrate over x, y, and z.
- Hollywood