00:01
A few general comments about finding the center of mass of a planar object.
00:08
So we're talking about a two -dimensional object.
00:12
The position of that center, and sometimes that is called the centroid, if your density is one, and you're really not talking about the mass of an object.
00:25
But the position of the center is the average found by integrative, creating position vector against the area of the object, the domain of its actual occupation and space.
00:43
And that area is a double integral.
00:47
If you're working with cartesian coordinates, the area is dx by d .y, which you should use if it's a planar type of object.
00:59
And this is going to involve three different integrals that you can do separately and then put them together.
01:09
So the bottom integral is just simply the x, d, y, and that could be a nested integral.
01:22
The limits go over the domain of the object.
01:30
And one of those could be nested, as we'll see, if your object is not rectangular.
01:38
Integral number two comes from the numerator, and that is putting in the x coordinate.
01:51
Again, there may be nesting that we'll discuss, and again, you do the integral over the domain of the object, and the third integral, of course, we're putting in the y coordinate, with, again, the integral being over the domain.
02:15
And because of the nesting, i think it's best to look at an example.
02:24
So we are going to find the centroid, given that we're looking at a region defined by this curve, y equals 1 over 3x plus 1.
02:37
And we're looking at the region under that curve up to x equals 5.
02:48
So i'll just kind of sketch that.
02:50
It is kind of a odd -looking region for an object.
02:56
Kind of look at airfoil or rocket wing.
03:02
So it starts over here at y -equals 1 on the y -axis, when xe -0, and it just kind of goes down, and we're going up to x -equals 5.
03:20
So let's see how we would set up our integrals, and this is where the nesting comes into play.
03:25
So we can either nest thinking of strips of the y variable going from zero to function of x, or we can think of the x variable going out to some inverse function of y.
03:57
So in the first part, y would be nested, and in the second part, x would be nested.
04:10
And we can see since we're given y as a function of x, it's probably easiest to choose the y as nested.
04:21
So our first integral is going to look like x going from 0 to 5 on the outside and y going from 0 to 1 over 3x plus 1.
04:40
And that is nested because its limits involve the outer variable.
04:49
And that just simply gives us 1 over 3x plus 1 minus 0.
05:04
Okay, and that looks like a logarithm.
05:06
So we will just say u is equal to 3x plus 1, d u over 3 equals d x so we basically have um logarithm of 3x plus 1 times a third ranging from x equal 0 to 5 so that that bottom integral is equal to let's just call that i1 the bottom is one 16 minus zero.
05:56
Now, if this were an object, this would be the mass, but we could just think about this as being the area of the region, the area occupied by the region.
06:15
Okay, so the next integral we are going to do, looks like we're going to integrate the x.
06:21
Sorry, no, the x, which is the outer integral.
06:28
Okay, so we're going to be doing the integral of x, dx on the outside, and y is going to range from 0 to 1 over 3x plus 1.
07:07
And we see that the inner integral is not going to be any different.
07:15
That will still just be y, ranging from 0 to that function.
07:20
So we're left with x over 3x plus 1.
07:25
And i'll show you how i would integrate this.
07:31
I would probably try to divide x into 3x plus 1.
07:40
Divide 3x plus 1 into x and we get a third with a remainder of minus 1 third.
08:23
So that's 1 over...