Explore an advanced mathematics lecture extending Archimedes' parabolic area formula to cubic functions. Delve into the historical significance of this calculus computation and its modern applications using projective geometry and Algebraic Calculus. Learn how the formula remarkably extends to cubics by reframing it to involve the unique point at infinity. Examine the limitations of this approach for higher-degree polynomials. Gain insights into a modern approach to curves and surfaces that combines historical perspectives with contemporary linear algebra and algebraic analysis techniques.