Explore the historical development and mathematical intricacies of integrating x^n in this 44-minute lecture from the "Famous Math Problems" series. Delve into the ancient roots of calculus, tracing the problem from Archimedes to Cavalieri and beyond. Examine the challenges in defining integrals, including issues with real numbers, infinite processes, and the theory of areas. Gain insights into a new "Algebraic Calculus" approach that addresses these foundational concerns. Learn about the parabola quadrature, Cavalieri's formula, and the connection to linear algebra's concept of area.