8
FURTHER APPLICATIONS
OF INTEGRATION
APPLICATIONS TO PHYSICS AND ENGINEERING
To work, our strategy is:





Break up the physical quantity into small parts.
Approximate each small part.
Add the results.
Take the limit.
Then, evaluate the resulting integral.
MOMENTS AND CENTERS OF MASS
Our main objective here is to find the point P
on which a thin plate of any given shape
balances horizontally as shown.
CENTERS OF MASS
This point is called the center of mass
(or center of gravity) of the plate.
CENTERS OF MASS
We first consider the simpler
situation illustrated here.
CENTERS OF MASS
Two masses m1 and m2 are attached to
a rod of negligible mass on opposite sides
of a fulcrum and at distances d1 and d2 from
the fulcrum.
CENTERS OF MASS
Equation 2
The rod will balance if:
m1d1  m2 d 2
LAW OF THE LEVER
This is an experimental fact discovered by
Archimedes and called the Law of the Lever.
 Think of a lighter person balancing a heavier one
on a seesaw by sitting farther away from the center.
MOMENTS AND CENTERS OF MASS
Now, suppose that the rod lies along
the x-axis, with m1 at x1 and m2 at x2
and the center of mass at
x.
MOMENTS AND CENTERS OF MASS
Comparing the figures,
we see that:
 d1 = x – x 1
 d2 = x1 – x
Equation 3
CENTERS OF MASS
So, Equation 2 gives:
m1 ( x  x1 )  m2 ( x2  x )
m1 x  m2 x  m1 x1  m2 x2
m1 x1  m2 x2
x
m1  m2
MOMENTS OF MASS
The numbers m1x1 and m2x2 are called
the moments of the masses m1 and m2
(with respect to the origin).
MOMENTS OF MASS
m1 x1  m2 x2
x
m1  m2
Equation 3 says that the center of mass x
is obtained by:
1. Adding the moments of the masses
2. Dividing by the total mass m = m1 + m2
CENTERS OF MASS
Equation 4
In general, suppose we have a system of n
particles with masses m1, m2, . . . , mn located
at the points x1, x2, . . . , xn on the x-axis.
Then, we can show where the center of mass
of the system is located—as follows.
Equation 4
CENTERS OF MASS
The center of mass of the system is
located at
n
x
n
m x m x
i 1
n
i i
m
i 1

i i
i 1
m
i
where m = Σ mi is the total mass of
the system.
MOMENT OF SYSTEM ABOUT ORIGIN
The sum of the individual moments
n
M   mi xi
i 1
is called the moment of the system
about the origin.
MOMENT OF SYSTEM ABOUT ORIGIN
Then, Equation 4 could be rewritten
as:
mx =M
 This means that, if the total mass were considered
as being concentrated at the center of mass x ,
then its moment would be the same as the moment
of the system.
MOMENTS AND CENTERS OF MASS
Now, we consider a system of n particles with
masses m1, m2, . . . , mn located at the points
(x1, y1), (x2, y2) . . . , (xn, yn) in the xy-plane.
Equations 5 and 6
MOMENT ABOUT AXES
By analogy with the one-dimensional case,
we define the moment of the system about
the y-axis as
n
M y   mi xi
i 1
and the moment of the system about
the x-axis as
n
M x   mi yi
i 1
MOMENT ABOUT AXES
My measures the tendency of the system
to rotate about the y-axis.
Mx measures the tendency of the system
to rotate about the x-axis.
Equation 7
CENTERS OF MASS
As in the one-dimensional case,
the coordinates ( x , y ) of the center of mass
are given in terms of the moments by
the formulas
x
My
m
Mx
y
m
where m = ∑ mi is the total mass.
MOMENTS AND CENTERS OF MASS
Since mx  M y and my  M x ,
the center of mass ( x , y ) is the point
where a single particle of mass m would
have the same moments as the system.
MOMENTS & CENTERS OF MASS
Example 3
Find the moments and center of mass
of the system of objects that have masses
3, 4, and 8 at the points (–1, 1), (2, –1) and
(3, 2) respectively.
MOMENTS & CENTERS OF MASS
Example 3
We use Equations 5 and 6 to compute
the moments:
M y  3(1)  4(2)  8(3)  29
M x  3(1)  4(1)  8(2)  15
MOMENTS & CENTERS OF MASS
Example 3
As m = 3 + 4 + 8 = 15, we use Equation 7
to obtain:
My
29
x

m 15
M x 15
y
 1
m 15
MOMENTS & CENTERS OF MASS
Example 3
Thus, the center of mass is:
14
15
(1 ,1)
CENTROIDS
Next, we consider a flat plate, called
a lamina, with uniform density ρ that
occupies a region R of the plane.
 We wish to locate the center of mass of the plate,
which is called the centroid of R .
CENTROIDS
In doing so, we use
the following physical
principles.
CENTROIDS
The symmetry principle says that, if R is
symmetric about a line l, then the centroid
of R lies on l.
 If R is reflected about l, then R remains the same
so its centroid remains fixed.
 However, the only fixed points lie on l.
 Thus, the centroid of a rectangle is its center.
CENTROIDS
Moments should be defined so that, if
the entire mass of a region is concentrated
at the center of mass, then its moments
remain unchanged.
CENTROIDS
Also, the moment of the union of two
non-overlapping regions should be
the sum of the moments of the individual
regions.
CENTROIDS
Suppose that the region R is of the type
shown here.
That is, R lies:
 Between the lines
x = a and x = b
 Above the x-axis
 Beneath the graph
of f, where f is a
continuous function
CENTROIDS
We divide the interval [a, b] into n subintervals
with endpoints x0, x1, . . . , xn and equal width
∆x.
 We choose the
sample point xi* to
be the midpoint xi
of the i th
subinterval.
 That is,
xi = (xi–1 + xi)/2
CENTROIDS
This determines the polygonal
approximation to R shown below.
CENTROIDS
The centroid of the i th approximating
rectangle Ri is its center Ci ( xi , 1 f ( xi )) .
2
Its area is:
f( xi ) ∆x
So, its mass is:
 f ( xi ) x
CENTROIDS
The moment of Ri about the y-axis is
the product of its mass and the distance
from Ci to the y-axis,
which is xi .
CENTROIDS
Therefore,
M y ( Ri )    f ( xi )x  xi
  xi f ( xi )x
CENTROIDS
Adding these moments, we
obtain the moment of the polygonal
approximation to R .
CENTROIDS
Then, by taking the limit as n → ∞,
we obtain the moment of R itself about
the y-axis:
n
M y  lim   xi f ( xi ) x
n 
b
i 1
   x f ( x) dx
a
CENTROIDS
Similarly, we compute the moment of Ri
about the x-axis as the product of its mass
and the distance from Ci to the x-axis:
M x ( Ri )    f ( xi )x  12 f ( xi )
 
1
2
 f ( xi )
2
x
CENTROIDS
Again, we add these moments and take
the limit to obtain the moment of R about
the x-axis:
n
M x  lim   
n 
 
b
i 1
1
a 2
1
2
 f ( xi )
2
 f ( x) 
2
dx
x
CENTROIDS
Just as for systems of particles,
the center of mass of the plate is
defined so that
mx  M y
and my  M x
CENTROIDS
However, the mass of the plate is
the product of its density and its area:
m  A
b
   f ( x) dx
a
CENTROIDS
Thus,
b
x
My
m

  xf ( x) dx
a
b
  f ( x) dx
a
xf ( x) dx


 f ( x) dx
M x    f ( x) dx
y


b
m
  f ( x) dx
b
1
a 2
a
2
b
a
b
a
  f ( x) dx
 f ( x) dx
b
2
1
a 2
b
a
 Notice the cancellation of the ρ’s.
 The location of the center of mass is independent
of the density.
Formula 8
CENTROIDS
In summary, the center of mass of the plate
(or the centroid of R ) is located at the point
( x , y ) , where
1 b
x   xf ( x) dx
A a
1 b1
2
y   2  f ( x)  dx
A a
CENTERS OF MASS
Example 4
Find the center of mass of a semicircular
plate of radius r.
 To use Equation 8, we place the semicircle as shown
so that f(x) = √(r2 – x2) and a = –r, b = r.
CENTERS OF MASS
Example 4
Here, there is no need to use the formula
to calculate
x.
 By the symmetry principle, the center of mass
must lie on the y-axis, so x  0 .
Example 4
CENTERS OF MASS
The area of the semicircle is A = ½πr2.
r
1
2
1
Thus, y 
f ( x)  dx

2

A r
2
1 1 r
2
2
 1 2 2
r  x dx
r

r
2


2
 2
r
2
0  r  x  dx   r 2
r
2
2
r
 2
x 
r x  3 

0
3
3
2 2r
4r
 2

 r 3 3
CENTERS OF MASS
Example 4
The center of mass is located at
the point (0, 4r/(3π)).
CENTERS OF MASS
Example 5
Find the centroid of the region bounded
by the curves
y = cos x, y = 0, x = 0, x = π/2
Example 5
CENTERS OF MASS
The area of the region is:
A
 2
0
cos x dx  sin x 0
 2
1
Example 5
CENTROIDS
So, Formulas 8 give:
1  /2
x   xf ( x) dx
A a

 /2
0
x cos x dx
 x sin x 0
 2

 2
0


2
1
sin x dx
1  /2 1
2
y   2  f ( x)  dx
A a
 /2

1
4


 1  cos 2 x  dx

1
4
x 

1
2
a
cos 2 x dx
/2
a


8
1
2
sin 2 x 0
 /2
CENTROIDS
Example 5
The centroid is ((π/2) – 1, π/8).
CENTROIDS
Suppose the region R lies between
two curves y = f(x) and y = g(x), where
f(x) ≥ g(x).
Formula 9
CENTROIDS
Then, the same sort of argument that led
to Formulas 8 can be used to show that
the centroid of R is ( x , y ) , where
1 b
x   x  f ( x)  g ( x)  dx
A a
1 b1
2
2
y   2  f ( x)    g ( x)  dx
A a


CENTROIDS
Example 6
Find the centroid of the region
bounded by the line x = y and
the parabola y = x2.
CENTROIDS
Example 6
The region is sketched here.
We take f(x) = x, g(x) = x2, a = 0, and b = 1
in Formulas 9.
Example 6
CENTROIDS
First, we note that the area of the region
is:
1
A   ( x  x ) dx
2
0
1
x
x 

 
2 2 0
1

6
2
3
Example 6
CENTROIDS
Therefore,


1 1
1 11
2
2
x   x  f ( x)  g ( x)  dx y  0 2  f ( x)   g ( x) dx
A
A 0
1 11 2
1 1
 1  2 ( x  x 4 ) dx
2
0
 1  x( x  x ) dx
6
6
0
1
 6  ( x  x ) dx
2
3
0
4 1
x x  1
 6   
 3 4 0 2
3
1
x x 
 3  
 3 5 0
2

5
3
5
CENTROIDS
The centroid is:  1 , 2 


2 5
Example 6