Hi, everyone!
Welcome back to integralcalc.com.
Today we’re going to be talking about how we’re going to find the centroid of a plane
region.
And in this particular problem, we’ve been asked to find the centroid of the plane regions
defined by these four lines, x = zero, x = 4, y = 0, and y = 6.
So let’s go ahead and really quickly just graph these since they’re easy to graph
and give us a visual of our plane region.
So we are looking at the lines x = 0, which is the line here across the y axis.
We’re looking at the line y = 0 which is this line here along the x axis.
And then we’re looking at x = 4 and y = 6.
So these are our four lines and this is therefore our plane region, this rectangle here.
and the centroid of a plane region is the coordinates of the geometrical center of the
region.
So in the case of a rectangle, we can pretty match eyeball that.
It’s halfway between obviously the vertical sides and it’s halfway between these horizontal
sides and if we draw diagonals through the center of the rectangle, we know that our
center is right about here.
so the centroid is going to be the coordinates of that point.
And the coordinates are written like these.
(x,y) with these lines over the top like that.
So we need to find the coordinates of that point.
And to find the x coordinate, we’re going to be using this first formula here.
To find the y coordinate, we’re going to be using the second formula.
And notice that in both formulas, we have a and a is defined by this third formula over
here.
Notice that we have a equals.
So the first thing we need to do is find a and then take that and plug it in to these
equations for the x and y coordinates for the centroid.
So to find a, we’ll say a is equal to the integral; notice that we have the integral
on the range a to b with an upper and lower limit.
This is a definite integral.
So you’re just looking for the left-hand point and the right-hand point.
And in this case, because we have a perfect rectangle, it’s really pretty easy.
We are looking for this point here which is the left most point of the region and this
point here which is the rightmost point of the region.
Since we have the lines x = 0 and x = 4, we know that this point here is at zero and this
one is at 4.
So we’re going to be evaluating on the range zero to 4.
And then we’re going to be evaluating f(x).
Well, f(x) is going to be the equation y = 6.
So we’re just going to plug in 6 over there and then of course we have our dx.
And the reason for that is because if these were another function, if we wanted to find
the area under a function, a normal function would be something like this, roughly parallel
to the x axis running horizontally like this and we would define the range 0 to 4 and then
we would take this graph here.
the closest graph we have to that is this line up here.
it’s not going to be one of these vertical lines, x = 0 and x = 4, and it’s not going
to be y = 0, so this line here y =6 is what we want to use for f(x).
We want to find the area underneath this line y =6 on the range zero to 4 that will give
us the area of our entire rectangle.
So that’s where we plug in 6 to this integral here and if we take the integral of 6, obviously
we’ll just get 6x and we’re going to be evaluating 6x on the range zero to 4.
So remember with definite integrals, when we evaluate on the range, we plug in the upper
limit first which is 4 so we get 6 times 4, then we subtract and plug in the lower limit
which in this case is zero which leaves us with 24 minus 0 which of course is just 24.
And that should make sense because since we’re dealing with a perfect rectangle, we have
this line y = 6 and we have this line x = 4 and the other two lines are along the axis
so we know that the area of this rectangle is 24 because we can multiply 6 times 4.
So we’ve kind of proven to ourselves that we can find the area of this rectangle using
this formula over here.
So the area is 24.
Now we can go ahead and plug it in to our formulas for the x and y coordinates.
So to find the x coordinate, we’ll get 1/24 by the formula here and then we’ll take
the integral again from zero to 4 of x which is part of the formula times f(x) and again
we’re going to call f(x) the equation y = 6.
This is going to be f(x) in every case, so we’re going to multiply by 6 and then we’ve
got dx.
So this ends up being 1/24 times the integral of 6x.
The integral of 6x is 3x squared and we’re going to be evaluating that on the range 0
to 4.
So let’s first go ahead and simplify 1/24 times 3x squared.
When we do, we’ll get x squared over 8 and this is now what we can easily evaluate on
the range 0 to 4.
Again we plug in 4 first, the upper limit and we’ll get 16/8 then we’ll subtract
and plug in zero, we get 0/8.
So that we can see that’s just going to be equal to 2.
This is going to become zero, 16/8 is just 2.
So the x coordinate is 2.
So we’ll go ahead and write that down.
And now we need to find the y coordinate.
So again, y = 1/24 times the integral from zero to 4.
1/2 is part of the formula so we leave that in there and then f(x) again will be 6 but
according to the formula, we need to square that.
So we square it.
dx is part of our notation.
So now we can go ahead and just solve this integral.
When we simplify what’s inside the integral, we get 36/2 which is just going to be 18 dx.
Taking the integral of 18 gives us 18x so we end up with 1/24 times 18x and then we’re
going to evaluate on the range 0 to 4.
We can simplify 1/24 times 18x.
We’ll get 3x/4.
Evaluate it on the range 0 to 4.
We’ll plug in our upper limit first.
So 4 times 3 gives us 12/4 minus 3 times 0 is 0/4.
This obviously becomes zero and 12/4 is equal to 3 so that is our y coordinate.
So the centroid of our plane region is this coordinate here which we get from the formula
and we know that that’s going to be equal to (2,3).
And that’s going to be our final answer.
And as a final point, this should make sense in the context of this particular problem
because our plane region is a rectangle.
So we already know the center.
If we go halfway between the x boundaries here zero and 4, halfway across would be 2.
So the x coordinate we already know to be 2.
And if we go from zero to 6 on the y axis, halfway would be 3.
So we could have eyeballed this just by looking at it and we would have known that the coordinate
of that point was (2,3).
But these are the formulas that you would use to prove it and certainly if you had a
more complicated problem, you would need these formulas.
But that’s it.
That’s our final answer.
I hope this video helped you guys and I will see you in the next one.
Bye!