00:01
Okay, we want to integrate x squared over the tetrahedron formed by these corner points, these vertices, so that's just, whoops, the standard tetrahedron in the first quadrant, first octant, whoops, okay.
00:21
Alright so it looks like that.
00:24
So if we're going to integrate z first, we're going to go from z equals zero, which is the xy plane, up to z equals this plane right here.
00:33
So we're going to find the equation at a plane.
00:35
Okay, so we can do that by just doing the cross product, we've got to write the equation on the plane, so we're going to figure out this vector and this vector, whoops, let me get a different color there, we'll figure out the name of this vector and this vector, and then we'll do their cross product.
01:01
Okay, so the first one, this vector right here would be minus one, zero, one, and then this one would be zero minus one, one, and so when we do the cross product, we have i, j, k, minus one, zero, one, zero minus one, one, so i times zero plus one minus j times negative one plus k times one, so one, one, one, so that's just going to be x plus y plus z equals d, and then d would be, just pick a point, any point on there, let's say zero, zero, one, plug it in, zero, zero, one, so x plus y plus z equals one...