00:01
So they want us to find the center of mass for this region here, given our constant density.
00:09
So for us to find the center, they give us these three equations over here.
00:18
So the first thing, notice that both of these share m.
00:22
So we can go ahead and calculate this first, and then go from there.
00:30
One of the things to kind of keep in mind, and this only applies when our density is a constant as opposed to varying, all of these deltas will end up canceling out with each other once we divide.
00:45
Because notice it's 1 over m, so this delta gets factor out, that delta gets factor out, and they cancel out with each other.
00:51
So when we're doing these calculations, and this is only because delta is constant, we can just cancel them all out like that and then not have to worry about it.
00:59
So first let's go ahead and figure out what our region here is.
01:05
So we can figure out what is f and what is g.
01:08
So if i were to sketch this really quickly, so x squared will look something like this, and then y is equal to 4 will be here.
01:25
So notice in this region, so this pretty much goes back to like when we were doing solids of revolutions.
01:33
And all we really want to do is look for our bottom function, look for our top function.
01:38
So in this case, i'm going straight up like this.
01:41
So notice the bottom is going to be the red, or in other words, this is g of x.
01:47
And then our top function is going to be the black, which is going to be f of x.
01:53
So let's go ahead and plug those in over here really quickly.
01:55
So this is going to be 4 minus x squared dx.
02:00
And now we need to figure out, well, what is a, what is b? well, that is going to be where these two functions intersect each other, so we can set them equal.
02:12
So we'd have x squared is equal to 4.
02:14
We'll just take the square root on each side, and we get x is equal to plus or minus 2.
02:18
So our bottom bound is 2, our upper bound is 2.
02:21
Our bottom bound is negative 2 and our upper bound is 2.
02:25
So let's go ahead and integrate this now.
02:29
And actually, one thing you might notice is that this is an even function.
02:35
Use the property since we're integrating on a symmetric interval to just multiply by the 2 and start from 0 like this.
02:43
Now the only reason why i want to do this is just to simplify the algebra layer.
02:46
It's not really needed.
02:48
All right.
02:48
So now let's go ahead and integrate.
02:50
So integrating this, so first we get 4x and then to integrate x squared, we're going to add 1 to the power, divide by the new power, and then evaluate from 0 to 2.
03:00
And now we can plug in two into here...