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:

- Rectangular section,
- Circular section, and
- 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=kx ^{2}*

*in which OA = 6 units.*

*and AB = 4 units.*

**Sol. **The equation of parabola is y = kx^{2} … (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

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