00:01
Now in this question, we have a hemisphere, right? and the hemisphere, let me try to draw this hemisphere, right? so the hemisphere is actually given by, it looks like something like this, right? homisphere, let me try to draw it like this, right? so let me try to draw it.
00:17
The hemisphere, right, like this.
00:19
And its base is in the xy plane, right? and then, you know, something like this.
00:26
So that would be the hemisphere, right? and the radius of this hemisphere is actually, according to the question, is actually nine, okay.
00:32
That's the origin of the s -y plane, right? so you can imagine this is and, of course, the other side is y, right? and this set is z, right? and you ask you to find the d -coordinate, the average c -coordinate of points in this hemisphere, right? well, the average d -compliance, of course, is given by integral, right, over the entire region of the hemisphere, right? i'm going to call it that mu -omega, and d -v is the volume, right, of this, and times, of course, the z, right? so that's integration we need to do.
01:01
And of course, because this is a hemisphere, so it's better to do it using spherical coordinates, right? so that would mean we would do it this way, right? d5, that's from 0 to 2 pi, right? and then you have ceta, dceta, sine ceta, and this is going from 0 to pi over 2, right, because it's a half sphere, right? sphosphorus.
01:26
And then you have to integrate, of course, d r squared, right? and that is going from 0 to 9, right? and then, of course, times z, z can be written as r cosine ceta, right? so this integration we have to do, right? so this part, of course, gives us 2 pi.
01:43
And this cosine theta, we can write it, of course, we can write it as, you know.
01:48
This is basically, this can be written as d, cossan ceta, basically...