6.5.1 Learning Goal: To be able to calculate the moment of inertia of composite areas. An object's moment of inertia is calculated analytically via integration, which involves dividing the object's area into the elemental strips that are parallel to the axes and then performing the integration of the strip's moment of inertia. In practice, engineers often encounter structural members that have areas composed of the common geometric shapes, such as rectangles, triangles, and circles, whose areas, centroidal locations, and moments of inertia about their centroidal axes are either known or can be calculated easily. Determining the moment of inertia of a composite area with respect to any axis is based on the following definitions: Iz Iy = Sy² dA Sæ² dA For a composite area consisting of a finite number of components, the integration is replaced by summation and the general equation simplifies to I = Σ²₁ I; where I; is the moment of inertia of the composite area's ith component with respect to the same reference axis. Therefore, the moment of inertia of a composite area with respect to a reference axis is equal to the algebraic sum of the moment of inertia of the components with respect to the same axis. In finding the moment of inertia of the components with respect to the desired axis, the parallel-axis theorem is sometimes necessary. Thus, I₂ = Īz + Ad² and Iy = Īv + Ad² where I and I are the moments of inertia of an area about its centroidal axes, A is the entire area, and dy and d are the perpendicular distances between the parallel axes. Compared to the integration method, this summation method is a simpler one for determining the moments of inertia of areas m ✦n →→ Part B - Moment of inertia of the composite area about the x axis The moment of inertia of the triangular shaped area is I₂ =1.00 x 105 mm². Given m = 70.0 mm and n = 35.0 mm, calculate the moment of inertia of the shaded area shown (Figure 1) about the x axis. Express your answer to three significant figures and include the appropriate units. ▸ View Available Hint(s) HA 1₂= Value n → Units