00:01
Okay, we want to integrate z over the volume enclosed by z equals x squared plus y squared and z equals 4.
00:07
Okay, so z equals, if you let y be zero, z equals x squared, that's this parabola.
00:15
And if you let x be zero, then z equals y squared is this parabola.
00:21
So we have this paraboloid here, and then it stops when it gets to z equals four.
00:32
Okay, so we're going to start at z equals the parabola, which is x squared plus y squared, up to z equals 4, z, d z.
00:44
I'm going to just put d a for the rest of it for now.
00:49
Okay, so to figure the rest of it, we just need to look in the xy plane.
00:55
So in the xy plane, we're looking at where x squared plus y squared equals 4.
01:02
Okay like you've squished this down okay so let's change to a cylinder or polar cylindrical right here so you can see that r is going from zero to two and then theta going zero to two pi because we got to get all the way around so zero to two pi r zero to two x squared plus y squared that's r squared 4 z, dz, r, d -r, d -theta.
01:47
That's d -a and polar, d -theta going 0 to 2 pi, r going 0 to 2 because this is a circle of radius 2, and then z going from the paraboloid, which is r squared up to 4.
02:05
Because if you drew any one of these little slivers up and down, you can see that they're bottom is the paraboloid, and the top is this.
02:15
Okay, so 0 to 2 pi, 0 to 2, integral of z, dz is z squared over 2, from r squared to 4, r, d, r, d, r, d, d, theta...