<< Chapter < Page Chapter >> Page >

Key equations

  • Mass of a lamina
    m = ρ a b f ( x ) d x
  • Moments of a lamina
    M x = ρ a b [ f ( x ) ] 2 2 d x and M y = ρ a b x f ( x ) d x
  • Center of mass of a lamina
    x = M y m and y = M x m

For the following exercises, calculate the center of mass for the collection of masses given.

m 1 = 2 at x 1 = 1 and m 2 = 4 at x 2 = 2

Got questions? Get instant answers now!

m 1 = 1 at x 1 = −1 and m 2 = 3 at x 2 = 2

5 4

Got questions? Get instant answers now!

m = 3 at x = 0 , 1 , 2 , 6

Got questions? Get instant answers now!

Unit masses at ( x , y ) = ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 )

( 2 3 , 2 3 )

Got questions? Get instant answers now!

m 1 = 1 at ( 1 , 0 ) and m 2 = 4 at ( 0 , 1 )

Got questions? Get instant answers now!

m 1 = 1 at ( 1 , 0 ) and m 2 = 3 at ( 2 , 2 )

( 7 4 , 3 2 )

Got questions? Get instant answers now!

For the following exercises, compute the center of mass x .

ρ = 1 for x ( −1 , 3 )

Got questions? Get instant answers now!

ρ = x 2 for x ( 0 , L )

3 L 4

Got questions? Get instant answers now!

ρ = 1 for x ( 0 , 1 ) and ρ = 2 for x ( 1 , 2 )

Got questions? Get instant answers now!

ρ = sin x for x ( 0 , π )

π 2

Got questions? Get instant answers now!

ρ = cos x for x ( 0 , π 2 )

Got questions? Get instant answers now!

ρ = e x for x ( 0 , 2 )

e 2 + 1 e 2 1

Got questions? Get instant answers now!

ρ = x 3 + x e x for x ( 0 , 1 )

Got questions? Get instant answers now!

ρ = x sin x for x ( 0 , π )

π 2 4 π

Got questions? Get instant answers now!

ρ = x for x ( 1 , 4 )

Got questions? Get instant answers now!

ρ = ln x for x ( 1 , e )

1 4 ( 1 + e 2 )

Got questions? Get instant answers now!

For the following exercises, compute the center of mass ( x , y ) . Use symmetry to help locate the center of mass whenever possible.

ρ = 7 in the square 0 x 1 , 0 y 1

Got questions? Get instant answers now!

ρ = 3 in the triangle with vertices ( 0 , 0 ) , ( a , 0 ) , and ( 0 , b )

( a 3 , b 3 )

Got questions? Get instant answers now!

ρ = 2 for the region bounded by y = cos ( x ) , y = cos ( x ) , x = π 2 , and x = π 2

Got questions? Get instant answers now!

For the following exercises, use a calculator to draw the region, then compute the center of mass ( x , y ) . Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by y = cos ( 2 x ) , x = π 4 , and x = π 4

( 0 , π 8 )

Got questions? Get instant answers now!

[T] The region between y = 2 x 2 , y = 0 , x = 0 , and x = 1

Got questions? Get instant answers now!

[T] The region between y = 5 4 x 2 and y = 5

( 0 , 3 )

Got questions? Get instant answers now!

[T] Region between y = x , y = ln ( x ) , x = 1 , and x = 4

Got questions? Get instant answers now!

[T] The region bounded by y = 0 , x 2 4 + y 2 9 = 1

( 0 , 4 π )

Got questions? Get instant answers now!

[T] The region bounded by y = 0 , x = 0 , and x 2 4 + y 2 9 = 1

Got questions? Get instant answers now!

[T] The region bounded by y = x 2 and y = x 4 in the first quadrant

( 5 8 , 1 3 )

Got questions? Get instant answers now!

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating y = m x around the x -axis between x = 0 and x = 1

Got questions? Get instant answers now!

Rotating y = m x around the y -axis between x = 0 and x = 1

m π 3

Got questions? Get instant answers now!

A general cone created by rotating a triangle with vertices ( 0 , 0 ) , ( a , 0 ) , and ( 0 , b ) around the y -axis. Does your answer agree with the volume of a cone?

Got questions? Get instant answers now!

A general cylinder created by rotating a rectangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( 0 , b ) , and ( a , b ) around the y -axis. Does your answer agree with the volume of a cylinder?

π a 2 b

Got questions? Get instant answers now!

A sphere created by rotating a semicircle with radius a around the y -axis. Does your answer agree with the volume of a sphere?

Got questions? Get instant answers now!

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid ( x , y ) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Quarter-circle: y = 1 x 2 , y = 0 , and x = 0

( 4 3 π , 4 3 π )

Got questions? Get instant answers now!

[T] Triangle: y = x , y = 2 x , and y = 0

Got questions? Get instant answers now!

[T] Lens: y = x 2 and y = x

( 1 2 , 2 5 )

Got questions? Get instant answers now!

[T] Ring: y 2 + x 2 = 1 and y 2 + x 2 = 4

Got questions? Get instant answers now!

[T] Half-ring: y 2 + x 2 = 1 , y 2 + x 2 = 4 , and y = 0

( 0 , 28 9 π )

Got questions? Get instant answers now!

Find the generalized center of mass in the sliver between y = x a and y = x b with a > b . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

Got questions? Get instant answers now!

Find the generalized center of mass between y = a 2 x 2 , x = 0 , and y = 0 . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

Center of mass: ( a 6 , 4 a 2 5 ) , volume: 2 π a 4 9

Got questions? Get instant answers now!

Find the generalized center of mass between y = b sin ( a x ) , x = 0 , and x = π a . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

Got questions? Get instant answers now!

Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x = b , b > 0 , and is rotated around the y -axis.

This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.

Volume: 2 π 2 a 2 ( b + a )

Got questions? Get instant answers now!

Find the center of mass ( x , y ) for a thin wire along the semicircle y = 1 x 2 with unit mass. ( Hint: Use the theorem of Pappus.)

Got questions? Get instant answers now!

Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
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
Practice Key Terms 6

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask