00:03
All right, for this problem, we are given a region e, which is the solid that lies under the paraboloid given here, and it's above the following disk in the x, y plane.
00:12
So we want to find the volume of e using cylindrical coordinates.
00:17
Okay.
00:18
So to do this, we are going to use the conversions for cylindrical coordinates, which is that x squared plus y squared equals r squared.
00:30
We have x equaling r cosine theta y equals r sine theta and we also have that d x d y d z will equal r d z d r d theta okay so those are are all of our conversions to cylindrical coordinates and what we'd like to see are the bounds so so in the xy plane, if you take your paraboloid and set z equal to zero, you'll get where the paraboloid intersects the xy plane.
01:17
You'll get x squared plus y squared equals four, which is actually a circle of radius two.
01:27
However, the disk that's provided here is the equation of a circle with radius one, but it's offset from the center.
01:37
So the radius we're looking at, or excuse me, the circle we're looking at is actually going to be in this region here.
01:48
So the z bounds are fairly straightforward because we're going from the x, y, plane, to the paraboloid.
01:55
But the radius and theta bounds are going to be a little more complicated.
01:59
So notice that the blue circle actually has no effect on our radius or theta bounds, because we're strictly inside this disk in the x, y, plane.
02:11
So this equation here is going to give us our theta and radius bounds.
02:17
So let's go ahead and do that substituting in x and y into that disk equation.
02:24
So we had x minus one squared plus y squared.
02:38
And we could put less than or equal to one.
02:42
Really we just need to look at the boundary, which is equal to one.
02:46
All right, so you can distribute on this left -hand side.
02:55
You should get r -squared cosine -squared theta minus 2 -r -cosin -theta plus 1, and then plus our r -squared sine -squared theta.
03:13
I'm going to factor out r -squared from the first and last terms on the left -hand side, and also subtract one from each side, which leaves you with a zero on the right hand side.
03:38
This is good because this cosine squared plus sine squared theta will equal 1.
03:43
So we've simplified our equation down to r squared minus 2r cosine theta equals 0.
03:54
And we can factor out r and this will give us our two bounds.
04:02
In fact, if you left the inequality, you actually would get the result that r must be positive and then r must be less than 2 cosine theta, which actually gives you specifically the relationship for the bounds is.
04:17
It doesn't matter much, but here is the result you should get from this.
04:22
So this will give us our radius bounds.
04:23
You notice it depends on theta.
04:26
Be very careful when you actually write out your theta bounds.
04:29
It would be very tempting to write from 0 to 2 pi because we see a full circle included.
04:34
However, you want to note that when theta, equals 0, you're at this point here.
04:42
And it turns out actually when theta is pi over 2 is actually when you get to the origin...