380 Higher Engineering Mathematics
38.5 Centroids
A lamina is a thin flat sheet having uniform thickness.
The centre of gravity of a lamina is the point where
it balances perfectly, i.e. the lamina’s centre of mass.
When dealing with an area (i.e. a lamina of negligible
thickness and mass) the term centre of area or centroid
is used for the point where the centre of gravity of a
lamina of that shape would lie.
If x and y denote the co-ordinates of the centroid C
of area A of Fig. 38.9, then:
x =
b
a
xy dx
b
a
y dx
and y =
1
2
b
a
y
2 dx
b
a
y dx
0
Area A
x 5 a
x 5 b
y
x
y 5 f(x)
y
x
C
Figure 38.9
Problem 7. Find the position of the centroid of
the area bounded by the curve y = 3x
2 , the x-axis
and the ordinates x = 0 and x = 2.
If (x , y) are co-ordinates of the centroid of the given
area then:
x =
2
0
x y dx
2
0
y dx
=
2
0
x(3x
2
) dx
2
0
3x
2 dx
=
2
0
3x
3 dx
2
0
3x
2 dx
=
3x 4
4
2
0
[x 3 ]
2
0
=
12
8
= 1.5
y =
1
2
2
0
y
2 dx
2
0
y dx
=
1
2
2
0
(3x
2
)
2 dx
8
=
1
2
2
0
9x
4 dx
8
=
9
2
x 5
5
2
0
8
=
9
2
32
5
8
=
18
5
= 3.6
Hence the centroid lies at (1.5, 3.6)
Problem 8. Determine the co-ordinates of
the centroid of the area lying between the curve
y = 5x − x 2 and the x-axis.
y = 5x − x 2 = x(5 − x). When y = 0, x = 0 or x = 5.
Hence the curve cuts the x-axis at 0 and 5 as shown
in Fig. 38.10. Let the co-ordinates of the centroid be
(x , y) then, by integration,
x =
5
0
x y dx
5
0
y dx
=
5
0
x(5x − x
2
) dx
5
0
(5x − x
2
) dx
=
5
0
(5x
2
− x
3
) dx
5
0
(5x − x
2
) dx
=
5x 3
3 −
x 4
4
5
0
5x 2
2 −
x 3
3
5
0
8
C
6
4
2
1
2
3
4
5
x
y
y 5 5x 2 x 2
y
x
0
Figure 38.10