00:01
Ok, so let's get started by setting up an integral to find the volume v.
00:05
Well, this one in spherical coordinates is going to be an integral with respect to the angle φ, so it's going to be from 0 to π.
00:15
Then we are going to have an integral with respect to the angle θ, from 0 to 2π.
00:21
Finally, we have an integral with respect to the radius, so this one is going to be from cosφ to 3.
00:28
And now we have the infinitesimal element of volume in spherical coordinates, which is ρ² sinφ, dρ dθ and dφ.
00:43
Perfect.
00:44
Ok, now the integral with respect to θ.
00:48
This is easy to compute.
00:51
The integral with respect to θ is just 2π.
00:54
So we are left with 2π multiplied by an integral from 0 to π, and now we can easily compute the integral with respect to ρ.
01:06
Well, this one is going to be ρ³ over 3, evaluated between cosφ and 3.
01:17
Perfect.
01:18
In...
01:19
Oh, multiplied by sinφ, obviously.
01:25
In dφ.
01:26
Perfect.
01:28
Ok, let's compute this integral here.
01:31
Well, here we have 2π multiplied by an integral from 0 to π of 9 sinφ in dφ minus 2π multiplied by cosθ of φ over 3 multiplied by sinφ...
