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Home Integral Theorems of Calculus Proof of the Fundamental Theorem  
See also: Fundamental Theorem of Calculus  
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Proof of the Fundamental Theorem
We conclude this section with a proof of the Fundamental Theorem of Calculus. PROOF (i) Let f(x) be the area under the curve y = f(t) from a to x, Imagine that the vertical line cutting the taxis at x moves to the right as in Figure 4.2.11. Figure 4.2.11 We show that the rate of change of F(x) is equal to the length f (x) of the moving vertical line. Suppose x increases by an infinitesimal amount ∆x > 0. Then is the area of an infinitely thin strip of width ∆x and height infinitely close to f(x). By the Rectangle Property the area of the strip is between the inscribed and circumscribed rectangles (Figure 4.2.12), m ∆x ≤ F(x + ∆x)  F(x) ≤ M ∆x. Figure 4.2.12 Dividing by ∆x, Since f is continuous at x, the values m and M are both infinitely close to f(x), and therefore The proof is similar when ∆x < 0. Hence F'(x) = f (x). PROOF (ii) Let F(x) be any antiderivative of f Then, by (i), In Section 3.7 on curve sketching, we saw that every function with derivative zero is constant. Thus for some constant C_{0}. Then so


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