00:01
Okay, i'm going to look at this sphere right here, which is x squared plus y squared plus z squared minus z.
00:12
We need to add 1 fourth to both sides.
00:22
So this actually becomes z minus 1 over 2, 1 fourth.
00:29
So it's a sphere with the center is at 0, 1 over 2, so it's over the xy plane.
00:47
So theta is going to go all around for sure, and then we need this angle right here, phi, and that comes from the cone, which is z is, well, rho cosine phi is z, and then rho sine phi is the square root of x squared plus y squared, so tangent of phi is 1, so phi is pi over 4, and then rho actually comes from the sphere, which is, we have this, so this is rho squared, and z is rho cosine phi, so actually rho is going to change from 0 to cosine of phi.
01:39
So the integral is 0 to 2 pi, 0 to pi over 4, and then rho is 0 to cosine of phi, rho squared sine phi, d rho, d phi, d theta.
01:58
Okay? and that's what we need to solve.
02:01
So from here, we get rho cubed over 3 sine of phi from 0 to cosine of phi, which comes out to be cosine cubed phi over 3, sine phi minus sine phi over 3, cosine of 0 is 1, and then now we're going to d phi this.
02:32
For that, this part right here, u is cosine of phi, so du becomes negative sine of phi, d phi, so it's going to be negative 1 over 3, u cubed, so u to the fourth over 4, minus, plus cosine of phi over 3 from 0 to pi over 4.
03:05
Okay, so if you put cosine to the pi over 4, it is root 2 over 3, fourth power would be 1 over 4, so negative 1 over 48 here, and then plus root 2 over 6 minus 0, so which is negative 1 over 12, plus 1 over 3.
03:45
So actually, we made a mistake here.
03:49
This is 0 here...