Algebraic Formula for the Area Under a Parabola

(The variables are the same as in "Area under a parabola".)

By the Fundamental Theorem of Calculus, if we can find a function Z such that dZ/dX = Y, then

Z(R) - Z(-R) = -R∫RY dx.

(Any such Z is called an anti-derivative of Y.) But if we know how to compute derivatives and anti-derivatives of polynomials, we know that when

Z = H*X - (H/R2)*X3/3,

then

dZ/dX = H - (H/R2)*X2= Y,

Plugging -R and R in for X, we get

Z(R) - Z(-R) = 4/3*H*R = 2/3*B*H

which is the formula we discovered in "Area under a parabola".