Direct cubic spline approximation to integrals with applications in nautical

Abstract A cubic spline is fitted directly to the integral function\documentclass{article}\pagestyle{empty}\begin{document}$$ F(x) = \int_a^x {f(t){\rm d}t} {\rm for }x \in [a,b] $$\end{document} where f is continuous in [ a , b ]. This method is applied, with the derivative boundary condition, to determine ship stability from a curve of righting levers (a curve relating a ship's righting lever to the angle of heel). This direct‐fit method is extended to two‐dimensional integrals and applied to obtain the buoyancy curve and hence the underwater ship volume.