Area & Line Elements Library
The geometric-properties tables as a live instrument: every area and arc element with its area, centroid, and moments of inertia about both the reference axes and the centroidal axes — under any of four orientations, since Ix, Iy, and the centroid coordinates all reassign when the shape rotates. Each result is verified on the spot by numeric integration.
Area Element
Line Element (Arc)
Arc with its centroid — always on the axis of symmetry, pulled toward the curve. FIG. E19-2
Orientation is bookkeeping with consequences: rotate any area 90° and Ix and Iy trade places while the centroid coordinates swap and change sign. The physics never changes — only which axis carries which number. That is why a joist is stiff on edge and floppy flat: same rectangle, swapped moments.