Aircraft Wing Torsional Polar Moment of Inertia (Aerospace & Mechanical Engineering)

An aerospace structural dynamics engineer at the UNLV Howard R. Hughes College of Engineering is designing a lightweight carbon-fiber composite internal spar bracket for an unmanned aerial vehicle (UAV).

The cross-section of the structural bracket occupies a planar wedge region $R$ in the $xy$-plane bounded below by the horizontal axis $y = 0$, on the right by the vertical root rib $x = 4$, and above by the aerodynamic camber curve $y = \sqrt{x}$ (with dimensions in millimeters).

To determine the component's torsional rigidity and resistance to high-speed aeroelastic flutter, the structural engineer must compute the second moment of area relative to the $y$-axis ($J_y$). The initial finite-element script formulates the integral in vertical chords (Type I):

To validate the code against numerical convergence errors and eliminate square-root expressions in intermediate steps:

1. Extract the original Type I integration boundaries for region $R$ and algebraically invert the boundary curve $y = \sqrt{x}$ to solve for $x$ in terms of $y$.
2. Determine the constant $y$-limits and functional $x$-limits required to re-slice the region as a Type II horizontal layout ($dx \, dy$).
3. Formulate and evaluate the switched iterated integral $J_y = \iint_R x^2 \, dA$, providing the exact fractional result in $\text{mm}^4$.
4. Explain how computing moments of inertia via switched integration bounds verifies structural resistance against torsional warping in aerospace airframes.