Mechanical Engineering Principles

FIRST AND SECOND MOMENT OF AREAS 85

(ii) First moment of area of shaded strip about

axisOY=(yδx)(x)=xyδx

Total first moment of areaPQRSabout

axisOy

=limit

δx→ 0

∑x=b

x=a

xyδx=

∫b

a

xydx

(iii) First moment of area of shaded strip about
axisOx

=(yδx)

(y

2

)

=

1

2

y^2 x

Total first moment of areaPQRSabout

axisOx

=limit

δx→ 0

x∑=b

x=a

1

2

y^2 δx=

1

2

∫b

a

y^2 dx

(iv) Area ofPQRS,A=

∫b

a

ydx

(see ‘Engineering Mathematics, 3RDEdition’,

page 448)

(v) Letxandybe the distances of the centroid

of areaAaboutOyandOxrespectively then:

(x)(A)=total first moment of areaA

about axisOy

=

∫b

a

xydx

from which, x=

∫b

a

xy dx

∫b

a

ydx

and (y)(A)=total moment of areaA

about axisOx=

1

2

∫b

a

y^2 dx

from which, y=

1

2

∫b

a

y^2 dx

∫b

a

ydx

7.4 Centroid of area between a curve

and they-axis

Ifxandyare the distances of the centroid of area

EFGHin Figure 7.3 fromOyandOxrespectively,

then, by similar reasoning as above:

(x)(total area)=limit

δy→ 0

y∑=d

y=c

xδy

(x

2

)

=

1

2

∫d

c

x^2 dy

0

H G

C(x− 2 ,y)

E

F

x= f(y)

y

x

x

y

y= c

y= d

dy

Figure 7.3

from which, x=

1

2

∫d

c

x^2 dy

∫d

c

xdy

and(y)(total area)=limit

δy→ 0

y∑=d

y=c

(xδy)y

=

∫d

c

xydy

from which, y=

∫d

c

xy dy

∫d

c

xdy