A NEW APPROACH TO BOUNDING THE PERIMETER OF AN ELLIPSE: EXTREMAL ANALYSIS AND INTEGRAL REFORMULATION

The perimeter of an ellipse is typically represented using an elliptic integral, which does not have a closed-form solution. One common approach to approximating the ellipse perimeter involves determining its bounds using arithmetic, geometric, and harmonic means. This study refines the integral formulation representing the ellipse perimeter as the average of its integral. By restructuring the integration limits, a new method is introduced to determine the bounds based on the extreme values of the integrand. This approach offers a clearer geometric interpretation, establishes new lower and upper bounds for the perimeter, and proposes a conjecture for the greatest possible lower bound, supported by analysis of the trapezoidal rule with a single subinterval.