00:01
Okay, we want to find the volume of the tetrahedron that's in the first octant, bounded above by the tetrahedron or by the plane x plus y plus z equals 1.
00:16
Okay, so we don't really need the picture, but it looks something like this.
00:20
Okay, so if we're going to double integrate, we're going to go from zero up to the plane, and the plane is z equals 1 minus x minus y so that's what we're going to integrate okay and then in the xy plane we have this and that's one to one and i know those are ones because when y and z are zero x is one and when x or zero y is one okay so if we integrate vertically first we're going from y equals 0 to y equals this red line and this red line is y equals mx plus b.
01:07
M is the slope and the slope is minus 1 and b is the y intercept and it's 1 so from 0 to minus x plus 1 d y and then we're going to stack those up from here x equals 0 to here x equals 1 okay so that's the integral so 0 to 1.
01:31
We're integrating with respect to y.
01:33
So we get y minus xy, minus y squared over 2 from 0 to, i'm going to call it 1 minus x here.
01:46
So 0 to 1, 1 minus x minus x minus x minus 1 minus x times 1 minus x minus 1 1⁄2 dx.
01:59
Let's go ahead and multiply that out.
02:01
0 to 1...