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Next, we need to find the total mass of the rectangle. Let ρ represent the density of the lamina (note that ρ is a constant). In this case, ρ is expressed in terms of mass per unit area. Thus, to find the total mass of the rectangle, we multiply the area of the rectangle by ρ . Then, the mass of the rectangle is given by ρ f ( x i * ) Δ x .

To get the approximate mass of the lamina, we add the masses of all the rectangles to get

m i = 1 n ρ f ( x i * ) Δ x .

This is a Riemann sum. Taking the limit as n gives the exact mass of the lamina:

m = lim n i = 1 n ρ f ( x i * ) Δ x = ρ a b f ( x ) d x .

Next, we calculate the moment of the lamina with respect to the x -axis. Returning to the representative rectangle, recall its center of mass is ( x i * , ( f ( x i * ) ) / 2 ) . Recall also that treating the rectangle as if it is a point mass located at the center of mass does not change the moment. Thus, the moment of the rectangle with respect to the x -axis is given by the mass of the rectangle, ρ f ( x i * ) Δ x , multiplied by the distance from the center of mass to the x -axis: ( f ( x i * ) ) / 2 . Therefore, the moment with respect to the x -axis of the rectangle is ρ ( [ f ( x i * ) ] 2 / 2 ) Δ x . Adding the moments of the rectangles and taking the limit of the resulting Riemann sum, we see that the moment of the lamina with respect to the x -axis is

M x = lim n i = 1 n ρ [ f ( x i * ) ] 2 2 Δ x = ρ a b [ f ( x ) ] 2 2 d x .

We derive the moment with respect to the y -axis similarly, noting that the distance from the center of mass of the rectangle to the y -axis is x i * . Then the moment of the lamina with respect to the y -axis is given by

M y = lim n i = 1 n ρ x i * f ( x i * ) Δ x = ρ a b x f ( x ) d x .

We find the coordinates of the center of mass by dividing the moments by the total mass to give x = M y / m and y = M x / m . If we look closely at the expressions for M x , M y , and m , we notice that the constant ρ cancels out when x and y are calculated.

We summarize these findings in the following theorem.

Center of mass of a thin plate in the xy -plane

Let R denote a region bounded above by the graph of a continuous function f ( x ) , below by the x -axis, and on the left and right by the lines x = a and x = b , respectively. Let ρ denote the density of the associated lamina. Then we can make the following statements:

  1. The mass of the lamina is
    m = ρ a b f ( x ) d x .
  2. The moments M x and M y of the lamina with respect to the x - and y -axes, respectively, are
    M x = ρ a b [ f ( x ) ] 2 2 d x and M y = ρ a b x f ( x ) d x .
  3. The coordinates of the center of mass ( x , y ) are
    x = M y m and y = M x m .

In the next example, we use this theorem to find the center of mass of a lamina.

Finding the center of mass of a lamina

Let R be the region bounded above by the graph of the function f ( x ) = x and below by the x -axis over the interval [ 0 , 4 ] . Find the centroid of the region.

The region is depicted in the following figure.

This figure is the graph of the curve f(x)=squareroot(x). It is an increasing curve in the first quadrant. Under the curve above the x-axis there is a shaded region. It starts at x=0 and is bounded to the right at x=4.
Finding the center of mass of a lamina.

Since we are only asked for the centroid of the region, rather than the mass or moments of the associated lamina, we know the density constant ρ cancels out of the calculations eventually. Therefore, for the sake of convenience, let’s assume ρ = 1 .

First, we need to calculate the total mass:

m = ρ a b f ( x ) d x = 0 4 x d x = 2 3 x 3 / 2 | 0 4 = 2 3 [ 8 0 ] = 16 3 .

Next, we compute the moments:

M x = ρ a b [ f ( x ) ] 2 2 d x = 0 4 x 2 d x = 1 4 x 2 | 0 4 = 4

and

M y = ρ a b x f ( x ) d x = 0 4 x x d x = 0 4 x 3 / 2 d x = 2 5 x 5 / 2 | 0 4 = 2 5 [ 32 0 ] = 64 5 .

Thus, we have

x = M y m = 64 / 5 16 / 3 = 64 5 · 3 16 = 12 5 and y = M x y = 4 16 / 3 = 4 · 3 16 = 3 4 .

The centroid of the region is ( 12 / 5 , 3 / 4 ) .

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Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg
Practice Key Terms 6

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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