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math 3

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1
Chapter 2: Multiple Integrals
Apart from finding the area of region, surface area and volume by using double integrals, its application can be extended further in finding the
mass
,
moment
,
center of mass
(
or centroid
)
,
moment inertia
for
lamina
.
Any flat object with negligible thickness is called a
lamina
. Some examples of lamina are shown as follows: (a)
Homogeneous Lamina (b) Non-homogeneous Lamina
A lamina with
regular shape
and made from same material is called
homogeneous
lamina, and its density function,
( , )
x y
is constant
k
.
A lamina with
irregular shape
is called
non-homogeneous
lamina, and its density function,
( , )
x y
is expressed in terms of
x
and
y
.
If a lamina with continuous density function
( , )
x y
occupies a region
R
in the
xy
- plane , its total mass
m
is given by:
, ( , )
R
Mass m x y dA
Application of Double Integrals Application of Double Integrals: (a) Calculating the Mass of a Lamina
2
Chapter 2: Multiple Integrals
Example 1
A lamina bounded by
x
-axis,
x
= 1 and the curve
2
y x
has density
( , )
x y x y
. Find its total mass.
Solution
, ( , )
R
Mass m x y dA
10 01 20 01 201 32015 220
2( )02225 42 15 41320
y x x x y x x x x x x x
x ydydx y xy dx x x x dx x x dx x x
Example 2
A triangular lamina with vertices
(0,0),(0,1)
and
(1,0)
has density
( , )
x y xy
. Find its total mass.
Solution
Equation of line:
y mx c
1 0, 10 1
slope m
and
y
-intercept,
c
= 1 Hence, the equation of line which connects the points (0,1) and (1,0) is given by:
1
y x
2
y x
R
3
Chapter 2: Multiple Integrals
, ( , )
R
Mass m x y dA
110 011 20 01 201 2013 2014 3 20
2( 1)02( 2 1)21221 22 4 3 21 1 2 102 4 3 2124
y x x x y y x x x y x x x x x x
xydydx xydx x xdx x x xdx x x xdx x x x
Moment
of mass of an object taken at a point (
x
,
y
), is the quantity of
distance
multiplies
its
mass
.
For a lamina with region
R
, the
moment taken about the
y
-axis
is given by:
( , )
y R
M x x y dA
For a lamina with region
R
, the
moment taken about the
x
-axis
is given by:
( , )
x R
M y x y dA
Application of Double Integrals: (b) Calculating the Moment of a Lamina
4
Chapter 2: Multiple Integrals
Example 1
A triangular lamina with vertices
(0,0),(0,1)
and
(1,0)
has density
( , )
x y xy
. Find its moment of mass about
x
-axis.
Solution
Moment of mass about
x
-axis:
( , )
x R
M y x y dA
1 10 011 30 01 301 3 201 4 3 2015 4 230
( )3( 1)3( 3 3 1)33 331 33 5 4 21 1 3 113 5 4 2160
x x
y xy dydx xydx x xdx x x x xdx x x x x
dx x x x x
A point in a lamina where it is in a state of
equilibrium
is called
center of mass
, which can be obtained by dividing the moment of mass about
y
-axis and
x
-axis by the mass.
Application of Double Integrals: (c) Calculating the Center of Mass of a Lamina

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