We derive the Fundamental Formula of the Algebraic Calculus, which allows us to define, and evaluate, the signed area of a polynomially parametrized curve, that is a de Casteljau Bezier curve. The approach is not via limits of Riemann sums, in fact infinite arguments do not appear. Rather we look at the crucial properties that linearly parametrized curves obey, and extending these to polynomially parametrized curves narrows down the possibilities and we get a unique formula.
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