TOPIC
APPLICATIONS
CENTRIODS OF PLANE AREA
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Determining Centroids of Plane Areas: Moments and Integration, Slides of Differential and Integral Calculus
Examples and formulas for calculating the centroids of plane areas using moments and integration. Topics include rectangular regions, parabolas, and representative strips. The document also includes formulas for moments of plane regions with respect to lines and coordinate axes.
Typology: Slides
2016/2017
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TOPIC
APPLICATIONS
CENTRIODS OF PLANE AREA
The mass of a physical body is a measure of the quantity of the matter in it,
whereas the volume of the body is a measure of the space it occupies.
If the mass per unit volume is the same throughout the body is said to
be homogeneous or to have constant density.
It is highly desirable in physics and mechanics to consider a given mass
as concentrated at a point, called its center of mass (also, its center of gravity).
For a homogeneous body, this point coincides with its geometric center or
centroid. For example, the center of mass of a homogeneous rubber ball
coincides with the centroid (center) of the ball considered as a geometric
solid (a sphere).
DISCUSSION
The centroid of a rectangular sheet of paper lies midway between the two
surfaces but it may well be considered as located on one of the surfaces
at the intersection of the two diagonals. Then the center of mass of a thin
sheet coincides with the centroid of the sheet considered as a plane area.
For a plane region having an area A, centroid and moments
and with respect to x and y axes,
C x , y ,
x
M y
M
y
M Ax x
M Ay
and
A
M
x
y
A
M
y
x
Example :
Determine the centroid of the first-quadrant region bounded by the parabola
.
2 y x
V(0,4)
y
x
dx
x x
C x y ,
Curve: 2
y 4 x
2
2
4
4
0,
x y
x y
V
y 0
2
0 4 x
2
4
2
x
x
if
Solving for the area A:
2
0
2
A ( 4 x ) dx
2
y y x
y
A B
y
2
y y x
A B
dA ( 4 x ) dx
2
2 2
0
2 2
0
2 2
0
2
2 2 2
0
2 2 4
0
2 3 5
0
sin 4
x x x x x x x x
M A y
M x y dx
y
but y
y
M x dx
x y dx
ce y x
M x x dx
M x x dx
x x
M x
M
M
^ ^ ^
x
M cu units
Moment about the x-axis
y
Moment about the y-axis
2 2 0 2 2 0 2 3 0 2
2 4
0
2
2 4
0
4
4
4
4
2 4
1
2
4
1
2 4 16
4
8 4
y y y y y y y y y
M A x
M x x dx
but x x
M x xdx
M x x dx
x x
M
M x x
M
M
M cu units
Example:
Determine the centroid of the fourth-quadrant area bounded by the curve
2
y x 4 x
.
V(2,-4)
dx
y
x
y
y
y
x x
Curve:
2
y x 4 x
2
2
4 4 4
2 4
2, 4
x x y
x y
V
0; 4 0
4 0
0; 4
y x x
x x
x x
4 0 2 4 2 0
4
4
dA y dx
A y dx
y x x
A x x dx
but
Solving for area A:
4 2 3 4 2
0
0
4 4
2 3
1 64
2 16 64 32
3 3
96 64 32
.
3 3
or
x x
A x x dx
A
A sq units
_
x
M A y
3
32
15
256
A
M
y
x
_
y units
5
8
units
64
3
32
3
2
y
y
M A x
M
x
A
x
^
5
8
C 2 ,
2
y x
y x
Example :
Determine the centroid of the region bounded by the curve
and the line.
.
dx
y
x
(x,y
C)
(x,y
L)
2 y x
y x
x x
y
y
Curve:
2 y x
V 0,0
y x
Line:
Intersection point:
2
2
0
1 0
0; 1
0; 0
1; 1
x x
x x
x x
x x
if x y
x y
1
0
1 2 2
0
(^1 ) 2 2
0
1 2 4
0
1 3 5
0
2 1 2 1 2 1 2 1
2 3 5
1 1 1
2 3 5
1 5 3
2 15
1
.
15
x
L C
x L C
x L C
x
x
x
x
x
M A y
y y
M y y dx
M y y dx
x x dx
M x x dx
x x
M
M
M
M cu units
^ ^
^ ^
1
0
1 2 2
0
1 2 2 2
0
1 2 4
0
1 3 5
0
2 1 2 1 2 1 2 1
2 3 5
1 1 1
2 3 5
1 5 3
2 15
1
.
15
x
L C
x L C
x L C
x
x
x
x
x
M A y
y y
M y y dx
M y y dx
x x dx
M x x dx
x x
M
M
M
M cu units
^ ^
^ ^
1
15
1
6
2
5
x
x
M A y
M
y
A
y units
1
12
1
6
1
2
y
y
M A x
M
x
A
x units
1 2
,
2 5
C
3 2
3 2
2 1
0
2 1 2 2 2 2 1 0
2
2
0 3 0 2 2 0 3 2 3
dA x x dy
A x x dy
x x y
x y
x y
x x y
A y y dy
y
A y dy
y
A y
A
A
A
A
sq units.
Solving for area A:
Continue solving for the centroid
Find the centroid of each of the given plane region bounded by the following curves:
1. y = 10x – x
2, the x-axis and the lines x = 2 and x = 5
- 2x + y = 6, the coordinate axes
- y = 2x + 1, x + y = 7, x = 8
- y2 = 2x, y = x – 4
- y = x3, y = 4x [first quadrant]
- y2 = x3, y = 2x
- y = x2 – 4, y = 2x – x
- the first quadrant area of the circle x2 +y2 = a
- the region enclosed by b2x2 + a2y2 = a2b2 in the first
quadrant