# FIRST MOMENT OF AREA AND CENTROID OF SECTION –RECTANGLE, CIRCLE, TRIANGLE, TRIANGLE FROM INTEGRATION

The first moment of area as defined in Art. 4.1 will be used to determine the centroid of the following sections by the method of integration:

1. Rectangular section,
2. Circular section, and
3. Triangular section

Centroid of a Rectangular Section by Integration.

Fig. 4.2 shows a rectangular section ABCD having width = b and depth = d. Consider a rectangular elementary strip of thickness ‘dy’ at a distance y from the axis OX.

Fig. 4.2

Fig. 4.2 (a)

Centroid of Circular Section by Integration.

Fig. 4.3

Fig. 4.3(a)

Centroid of a Triangular Section by Integration.

Problems of Finding Centre of Gravity of Areas by Integrations Method

Problem 4.1. Determine the co-ordinates of the C.G. of the area OAB shown in Fig. 4.5, if the curve OB represents the equation of a parabola, given by

y=kx2

in which OA = 6 units.

and AB = 4 units.

Sol. The equation of parabola is y = kx2                                        … (i)

First determine the value of constant k. The point B is lying on the curve and having co-ordinates

x = 6 and y = 4