quality:corruption, linguistic:translation, structural:scholar_edition, linguistic:voice_change
Proof

The first fundamental theorem may be interpreted as follows. Given a continuous function y = f (x) {\displaystyle y=f(x)} whose graph is plotted as a curve, one defines a corresponding "area function" x ↦ A (x) {\displaystyle x\mapsto A(x)} such that A (x) is the area beneath the curve between 0 and x. The area A (x) may not be easily computable, but it is assumed to be well defined.

The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this "strip" would be A (x + h) − A (x).

There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f (x) to find the area of a rectangle that is approximately the same size as this strip. So: A (x + h) − A (x) ≈ f (x) ⋅ h {\displaystyle A(x+h)-A(x)\approx f(x)\cdot h}

Dividing by h on both sides, we get: A (x + h) − A (x) h ≈ f (x) {\displaystyle {\frac {A(x+h)-A(x)}{h}}\ \approx f(x)}

This estimate becomes a perfect equality when h approaches 0: f (x) = lim h → 0 A (x + h) − A (x) h = def A ′ (x). {\displaystyle f(x)=\lim _{h\to 0}{\frac {A(x+h)-A(x)}{h}}\ {\stackrel {\text{def}}{=}}\ A'(x).} That is, the derivative of the area function A (x) exists and is equal to the original function f (x), so the area function is an antiderivative of the original function.

Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are inverse operations which reverse each other. This is the essence of the Fundamental Theorem.