00:01
Okay, so first of all, let me denote by omega, our hemisphere.
00:10
Okay, now the average z coordinate, z -a, is an integral is 1 over the measure of omega, multiplied by an integral over omega of z in the v, the infinitesimal element of volume.
00:35
Okay, so here we're going to use a spherical coordinates to compute this integral here.
00:44
So let me rewrite this integral.
00:48
Okay.
00:52
Well, in spherical coordinates, z is going to be actually maybe let me find the end points of integration first.
01:02
Then we are going to transform z in spherical coordinates.
01:06
Okay, the end points of integration are going to be from 0 to pi, the integral with respect, which is going to be with respect to fee.
01:18
Then we are going to have an integral from, this one is going to be an integral from 0 to pi over 2 with respect to theta, and an integral from 0 to 4, the radius.
01:35
And now we can transform z.
01:38
Well z is easy to transform.
01:41
Z is just raw multiplied by cosine of theta...