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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!