This document provides information about multiple integrals and examples of evaluating them. It begins by defining multiple integrals and iterated integrals. It then gives 4 examples of evaluating double and triple integrals over different regions. These regions include rectangles, triangles, and solids. The document also discusses Fubini's theorem, which allows reversing the order of integration in certain cases. It concludes by providing an example of converting an integral from Cartesian to polar coordinates.
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rahimahj@ump.edu.my
24. rahimahj@ump.edu.my
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37. A lamina is a flat sheet (or plate) that is so thin as to
be considered two-dimensional.
Suppose the lamina occupies a region D of the xy-
plane and its density (in units of mass per area) at a
point (x, y) in D is given by ρ(x, y), where ρ is a
continuous function on D. This means that
A
m
yx
lim),(
where Δm and ΔA are the mass and area of a small
rectangle that contains (x, y) and the limit is taken as
the dimensions of the rectangle approach 0.
Laminas & Density
39. rahimahj@ump.edu.my
1 1
0 0
11
2
00
A triangular lamina with vertices 0,0 , 0,1
and 1,0 has density function , .
Find its total mass.
Solution :
,
1 1
...
2 24
x
R
x
x y xy
m x y dA xy dydx
m xy dx unit of mass
Example 7
40. rahimahj@ump.edu.my
The moment of a point about an axis is the product of
its mass and its distance from the axis.
To find the moments of a lamina about the x- and y-
axes, we partition D into small rectangles and assume
the entire mass of each subrectangle is concentrated
at an interior point. Then the moment of Rij about the
x-axis is given by
and the moment of Rk about the y-axis is given by
****
),())(mass( ijijijij yAyxy
****
),())(mass( ijijijij xAyxx
Moment
41. rahimahj@ump.edu.my
m
i
n
j D
ijijij
nm
x dAyxyAyxyM
1 1
***
,
),(),(lim
The moment about the x-axis of the entire
lamina is
The moment about the y-axis of the entire
lamina is
m
i
n
j D
ijijij
nm
y dAyxxAyxxM
1 1
***
,
),(),(lim
42. rahimahj@ump.edu.my
Center of Mass
The center of mass of a lamina is the “balance point.”
That is, the place where you could balance the lamina
on a “pencil point.” The coordinates (x, y) of the center
of mass of a lamina occupying the region D and having
density function ρ(x, y) is
where the mass m is given by
D
x
D
y
dAyxy
mm
M
ydAyxx
mm
M
x ),(
1
),(
1
D
dAyxm ),(
45. 2
Find the mass and center of mass of the lamina
that occupies the region and has the given
density of function .
a) , 0 2, 1 1 ; ,
4 4
: , ,0
3 3
) is bounded by , 0, 0, and 1;x
D
D x y x y x y xy
Ans
b D y e y x x
x
32
2
2 2
,
4 11 1
: 1 , ,
4 2 1 9 1
y y
ee
Ans e
e e
Example 8
46. rahimahj@ump.edu.my
Example 9
Find the surface area of the portion of the surface
that lies above the rectangle R in the xy-
plane whose coordinates satisfy
2
4 xz
40and10 yx