1. Moments of inertia for areas
Previously determined centroid for an area by considering first moment of area about an axis,
Moment of inertia → second moment of an area (“moment of inertia” is a misnomer)
Moments of inertia of a differential planar area dA about the x and y axes are dIx = y2dA and dIy = x2dA, respectively SHOW
For the entire area,
Second moment of differential area dA about the pole O or z axis is called the polar moment of inertia, dJo = r2dA

2. Parallel-Axis Theorem
SHOW
However, x’ passes through the centroid of the area
Similarly,

3. Radius of Gyration of an Area
A quantity often used in the design of columns
(units are length)

4. Moments of inertia for an area by integration (Case 1)
Specify the differential element dA (usually a rectangle with finite length and differential width)
Differential element located so it intersects the boundary of the area at an arbitrary point (x,y)
Case 1:
Length of element is oriented parallel to axis, SHOW
Iy determined by a direct application of since element has an infinitesimal thickness dx (all parts of the element lie at the same moment-arm distance x from the y-axis)

5. Moments of inertia for an area by integration (Case 2)
Length of element is oriented perpendicular to axis, SHOW
does not apply, all parts of the element will not lie at the same moment-arm distance from axis
First calculate moment of inertia of the element about a horizontal axis passing through the element’s centroid
Determine moment of inertia of the element about the x-axis by using parallel-axis theorem
Integrate to yield Ix
EXAMPLES (pg 537)

6. Moments of inertia for composite areas
Divide the area into its “simpler” composite parts
Determine the perpendicular distance from the centroid of each part to the reference axis (often centroid of entire area)
Determine the moment of inertia of each part about its centroidal axis (parallel to the reference axis)
If the centroidal axis does not coincide with the reference axis, apply the parallel-axis theorem to determine the moment of inertia of the part about the reference axis
The moment of inertia of the entire area about the reference axis is obtained by summing the results of its parts (“holes” are subtracted)
EXAMPLES (pg 544)