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Centroids
Let
A
be a geometric figure in Euclidean space.
The Euclidean space may be
R
2
or
R
3
.
It could in fact be
R
n
.
The figure may be any dimension.
The coordinates of the
centroid
or
center of mass
of the figure are given by the following formulas.
x
=
x
!
dA
"
dA
"
y
=
y
!
dA
"
dA
"
z
=
z
!
dA
"
dA
"
Of course, if the figure is in
R
2
, only the first two coordinates apply.
The
dA
in the
formulas are small strips of the figure that are perpendicular to the direction of the
respective
x
–axis or
y
–axis or
z
–axis.
The simplest way to understand this is to work
through a few examples.
1.
Determine the centroid of the portion of the area of a disk of radius
R
centered at
the origin that is in the first quadrant.
x
=
x
!
R
2
"
x
2
dx
0
R
#
R
2
"
x
2
dx
0
R
#
=
4
R
3
$
y
=
y
!
R
2
"
y
2
dx
0
R
#
R
2
"
y
2
dx
0
R
#
=
4
R
3
2.
Determine the centroid of the area under the curve
f
(
x
)
=
x
n
over the interval
[0,1]
.
x
=
x
!
x
n
dx
0
1
"
x
n
dx
0
1
"
=
n
+
1
n
+
2
y
=
y
!
(1
#
y
n
)
dy
0
1
"
#
y
n
)
dy
0
1
"
=
n
+
1
4
n
+
2
x
n
R
2
!
x
2
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.
 Spring '08
 BLOCK
 Calculus

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