# 6.6 Moments and centers of mass  (Page 7/14)

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## Theorem of pappus

This section ends with a discussion of the theorem of Pappus for volume    , which allows us to find the volume of particular kinds of solids by using the centroid. (There is also a theorem of Pappus for surface area, but it is much less useful than the theorem for volume.)

## Theorem of pappus for volume

Let R be a region in the plane and let l be a line in the plane that does not intersect R . Then the volume of the solid of revolution formed by revolving R around l is equal to the area of R multiplied by the distance d traveled by the centroid of R.

## Proof

We can prove the case when the region is bounded above by the graph of a function $f\left(x\right)$ and below by the graph of a function $g\left(x\right)$ over an interval $\left[a,b\right],$ and for which the axis of revolution is the y -axis. In this case, the area of the region is $A={\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx.$ Since the axis of rotation is the y -axis, the distance traveled by the centroid of the region depends only on the x -coordinate of the centroid, $\stackrel{–}{x},$ which is

$\stackrel{–}{x}=\frac{{M}_{y}}{m},$

where

$m=\rho {\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{M}_{y}=\rho {\int }_{a}^{b}x\left[f\left(x\right)-g\left(x\right)\right]dx.$

Then,

$d=2\pi \frac{\rho {\int }_{a}^{b}x\left[f\left(x\right)-g\left(x\right)\right]dx}{\rho {\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx}$

and thus

$d·A=2\pi {\int }_{a}^{b}x\left[f\left(x\right)-g\left(x\right)\right]dx.$

However, using the method of cylindrical shells, we have

$V=2\pi {\int }_{a}^{b}x\left[f\left(x\right)-g\left(x\right)\right]dx.$

So,

$V=d·A$

and the proof is complete.

## Using the theorem of pappus for volume

Let R be a circle of radius 2 centered at $\left(4,0\right).$ Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y -axis.

The region and torus are depicted in the following figure.

The region R is a circle of radius 2, so the area of R is $A=4\pi$ units 2 . By the symmetry principle, the centroid of R is the center of the circle. The centroid travels around the y -axis in a circular path of radius 4, so the centroid travels $d=8\pi$ units. Then, the volume of the torus is $A·d=32{\pi }^{2}$ units 3 .

Let R be a circle of radius 1 centered at $\left(3,0\right).$ Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y -axis.

$6{\pi }^{2}$ units 3

## Key concepts

• Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
• For point masses distributed along a number line, the moment of the system with respect to the origin is $M=\sum _{i=1}^{n}{m}_{i}{x}_{i}.$ For point masses distributed in a plane, the moments of the system with respect to the x - and y -axes, respectively, are ${M}_{x}=\sum _{i=1}^{n}{m}_{i}{y}_{i}$ and ${M}_{y}=\sum _{i=1}^{n}{m}_{i}{x}_{i},$ respectively.
• For a lamina bounded above by a function $f\left(x\right),$ the moments of the system with respect to the x - and y -axes, respectively, are ${M}_{x}=\rho {\int }_{a}^{b}\frac{{\left[f\left(x\right)\right]}^{2}}{2}dx$ and ${M}_{y}=\rho {\int }_{a}^{b}xf\left(x\right)dx.$
• The x - and y -coordinates of the center of mass can be found by dividing the moments around the y -axis and around the x -axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
• The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.

why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
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
y=800
800
Bg
how do u factor the numerator?
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
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?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
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
is x=2 a function?
The
What is limit
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'
what is implicit of y³+x²y³=5 at (2,1)
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