In this chapter, we present the Archimedean method for calculating the area of a parabolic segment. This ingenious approach represents one of the earliest contributions to the theory of integration, predating its systematic development by nearly eighteen centuries. We show that the method can be readily generalized to curves of higher degree. Cubic, quartic, and quintic polynomials are treated in essentially the same manner as Archimedes’ original parabolic segment, requiring no major conceptual modifications. In this way, we highlight both the power of the Archimedean approach and its largely untapped potential.