Centroids of Plane Regions (KristaKingMath)

Centroids of Plane Regions (KristaKingMath)


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!

17 comments on “Centroids of Plane Regions (KristaKingMath)

  1. @SuCKeRPunCH187 For the most part. Some of the stuff I cover are topics that would be covered in the third or fourth class in a calculus series, but obviously all classes are different. 🙂

  2. HI, I stuck on a few questions similar to this example. I am able to work out the bounded area, aswell as the co ords of the centroid, however the third part of my problems are asking for the second moment of area about the x axis and the y axis…any idea how I should go about this???thaanks

  3. "Centroids" – that's a new one for me. In a 10 minute video on a new subject there's going to be some 'loose ends', but this is a great intro to the concept. Thank you Krista.

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