The Fundamental Theorem of Calculus

Integration is introduced as the reversal of differentiation i.e. in solving a differential equation, \dfrac{\textnormal{d}y}{\textnormal{d}x}=\textnormal{g}(x). The link between integration and area is often passed over and is the subject of the Fundamental Theorem of Calculus. [The following discussion can be adapted for a decreasing function or, piece-wise, a function which successively increases or decreases.]

Consider and area function, A(x), defined by the area under \textnormal{f}(x) between a and and a general point, x. If a small increment, \delta x, is applied to x giving a small element, \delta A of area. Now,

\textnormal{f}(x)\delta x \leqslant \delta A \leqslant \textnormal{f}(x+\delta x)\delta x

dividing though by \delta x, gives,

\textnormal{f}(x) \leqslant \dfrac{\delta A}{\delta x} \leqslant \textnormal{f}(x+\delta x),

a limit sandwich where, as \delta x \rightarrow 0,


The curve function, \textnormal{f}(x) is the derivative of the area function; hence the area function is the anti-derivative of the curve function and,

\displaystyle\int \textnormal{f}(x) \textnormal{d}x=A(x).


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