Analytic Parametric Multi-Step Solution of All Area and Moments Integrals of General Green’s Theorem for Arbitrary Ellipse Region, Part 2: Segment

In this Part 2 of the paper, analytical, parametric multi-step solution is given of all of the six area and moments integrals of arbitrary segment of ellipse defined in parametrical form by trigonometric functions, based on its corresponding integrals of its central sector from Part 1 of the paper. Those integrals are here built from the general moment form of Green’s theorem curve integral for the calculation of the area and moments of a planar shape enclosed by arbitrary curve, here applied for the ellipse segment calculations. The calculation for given central polar angle is then enabled by a relation between the circular parameter of a Cartesian, parametric representation of ellipse and a central polar angle given in Part 1. In the final calculation step, the area and moment values of the arbitrary segment of ellipse are calculated by using Boolean algebra applied on observed segment and its remaining triangle of the central ellipse sector. In this way, it is possible to analytically calculate general integrals of arbitrary ellipse segment for: area itself, its static moments, area centroids and moments of inertia of the central ellipse arc area, defined parametrically, by trigonometric functions. Additionally, this solution can be applied in the hydrostatics for the problem of determining the filling heights of horizontally symmetric segments of the elliptical liquid tank, together with its belonging center of buoyancy, being centroid of observed segment of ellipse.