Why does the area moment of inertia about the y-axis use $x^2$ in the integral?

Conditions

• The moment is computed over a planar area in the xy-plane.
• The axis of interest is either the x-axis or the y-axis.
• The distance is measured perpendicularly from the axis to the differential area element.

Reasoning, step by step

1. Identify the axis about which the moment of inertia is being calculated.
2. Determine the coordinate that represents the perpendicular distance from that axis to a generic point in the area.
3. Square this perpendicular distance coordinate.
4. Integrate this squared distance multiplied by the differential area element $dA$ over the entire region.

Example

The instructor explicitly states the rule: "if you're doing it about the y axis you have to find the x distance" and "if you're finding about the x axis you have to find the y distance." This is reflected in the board formulas $I_{x'} = \int y^2 dA$ and $I_{y'} = \int x^2 dA$.

Common misconceptions

• Using the coordinate parallel to the axis instead of the perpendicular distance (e.g., using $y^2$ for $I_y$).
• Assuming the prime notation on $I_{y'}$ changes the fundamental definition of the integrand rather than just indicating the axis location.

Watch the explanation

Explore next

• How do you set up the integral for the area moment of inertia about the y-axis using a vertical strip?
• Why does the area moment of inertia have units of $m^4$?
• When can you skip the parallel axis theorem in this setup?