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

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.