How to evaluate the triple integral where is the solid tetrahedon with vertices ?
Here's a similar problem although the points in that post aren't showing up but still it's the same: you first have to calculate a normal to the plane you're integrating under, then derive the equation of the plane, then integrate over the appropriate region in the x-y plane.
Maybe we should work this one through since it's been asked a few times. Using the plot below which is a little hard to visualize unless you can rotate it, I'll calculate two vectors: One goes from the point at (0,3,0) to (0,0,2) and the other goes from (8,0,0) to (0,3,0):
In order to calculate a normal to the blue surface, I calculate the cross-product of these two vectors which I've indicated as the line eminating from the origin.
I can now derive the equation of the blue surface using the equation of the plane passing through the point (0,3,0) and the normal vector:
That line in the back is right?
So then the integral is:
Pretty sure anyway.