The area moment of inertia of the following sections will be determined by the method of integration:

- Moment of inertia of a rectangular section,
- Moment of inertia of a circular section,
- Moment of inertia of a triangular section,
- Moment of inertia of a uniform thin rod.

**4.8.1 Moment of Inertia of a Rectangular Section**

**1 ^{st} Case. Moment of inertia of the rectangular section about X-X axis passing through the C.G. of the section.**

Fig. 4.22 shows a rectangular section ABCD having width = b and depth = d. Let X-X is the horizontal axis passing through the C.G. of the rectangular section. We want to determine the moment of inertia of the rectangular section about X-X axis. The moment of inertia of the given section about X-X axis is represent by I_{XX}.

Consider a rectangular elementary strip of thickness dy at a distance y from the X-X axis as shown in Fig. 4.22.

**2 ^{nd} Case. Moment of inertia of the rectangular section about a line passing through the base.**

Fig. 4.23 shows a rectangular section ABCD having width = b and depth = d. We want to find the moment of inertia of the rectangular section about the line CD, which is the base of the rectangular section.

Let X-X is the axis passing through the C.G. of the triangular section and parallel to the base.

The distance between the C.G. of the triangular section and base AB=h/3.

Now, from the theorem of parallel axis, given by equation (4.8), we have

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