00:01
All right, we want to find the volume of the tetrahedron in the first octant.
00:07
And the coordinate planes are three of the sides, and the other is this slanted plane.
00:13
So when y and z or x is three, when x and z are zero, y is two, and then when x and y are zero, z is six.
00:28
All right, so here's the plane.
00:30
Oops.
00:35
All right, so there's a tetrahedron.
00:37
Okay, you can do two or three integrals.
00:43
It turns out the same, of course, because you're finding the volume, but it even looks the same after the first step.
00:48
Okay, so first let's do z.
00:53
And so z is going from z equals zero, the x, y plane up to this plane, which is 6 minus 2x minus 3y.
01:04
So 6 minus 2x minus 3y minus 0, dz.
01:10
Oh, no, we did the z already.
01:12
Sorry okay and then now we have to look at the what we have in the xy plane okay we have this which is 2x plus 3 y equals 6 from here okay or y equals slope is negative 2 thirds down 2 over 3 plus the y intercept is 2 so x a y is going from 0 to that line 2 3 3 3 3 3 3x is going from 0 to that line 2 3 3 3 3 plus two, and then x is going zero to three...