Submersible Pressure Hull Buoyancy and Hydrostatic Stability (Mechanical & Ocean Engineering)

A mechanical and ocean systems engineer at the UNLV Howard R. Hughes College of Engineering is modeling the hydrodynamic stability of an unmanned underwater vehicle (UUV) designed for deep-sea trench exploration.

The symmetric profile of the submerged starboard hull section is bounded on the left by the vertical keel centerline $x = 0$, at the top by the design waterline $y = 3$, and on the exterior by the logarithmic hull boundary $x = \ln(y)$ (for $y \ge 1$, dimensions in meters). The static hydrostatic buoyant pressure moment is initially modeled as a Type II horizontal integral:

To integrate this profile directly into the automated hydrostatics solver and calculate the vertical center of buoyancy:

1. Extract the given Type II integration boundaries and invert the outer boundary function $x = \ln(y)$ to solve for $y$ explicitly in terms of $x$.
2. Determine the constant outer $x$-limits and functional inner $y$-limits required to re-slice the hull domain as a Type I vertical layout ($dy \, dx$).
3. Set up and evaluate the switched iterated integral $M = \iint_R y \, dA$, giving both the exact analytical expression and the rounded value to three decimal places.
4. Explain how computing moments of area via order-switched integrals aids ocean engineers in verifying transverse metacentric height and righting arm buoyancy stability.