Question

Calculate the second moment of area of a 4-in-diameter shaft about the x-x and yy axes, as shown.

Answer 1

Second moment of area about an axis along any diameter in the plane of the cross section (i.e. x-x, y-y) is each equal to (1/4)pi r^4.
The second moment of area about the zz-axis (along the axis of the cylinder) is the sum of the two, namely (1/2)pi r^4.

The derivation is by integration of the following:
int int y^2 dA
over the area of the cross section, and can be found in any book on mechanics of materials.