Vector Analysis

Fundamental Theorem of Calculus Top of Page Contents

Part I: If f(x) is continuous and F(x) = ∫₀^x f(t) dt, then F'(x) = f(x).

Proof.

F(x + h) - F(x) = ∫₀^x+h f(t)dt - ∫₀^x f(t)dt = ∫₀^x f(t)dt + ∫_x^x+hf(t) dt - ∫₀^x f(t) dt = ∫_x^x+h f(t) dt = f(x) h ,

F(x + h) - F(x)/h = f(x), \] \[ \lim_{h \to 0} \frac{F(x + h) - F(x)}{h} = f(x), \] \[ F'(x) = f(x). \]

Part II: If f is continuous and G is an antiderivative of f (i.e. G'(x) = f(x)), then ∫_a^b f(x) dx = G(b) - G(a).

Proof. \[ \int_a^b f(x) dx = \int_0^bf(x)dx - \int_0^a f(x) dx = F(b) - F(a). \]

Let be an antiderivative of , so that . \[ (F - G)'(x) = F'(x) - G'(x) = f(x) - f(x) = 0 \Rightarrow G(x) = F(x) + C, \] where C is a constant. \[ \int_a^b f(x) dx = F(b) - F(a) = (G(b) + C) - (G(a) + C), \] \[ \int_a^b f(x) dx = G(b) - G(a). \]

Consider a function of one variable, f(x). We saw in our section on integration that the area under the curve f(x) can be written as a Riemann integral. This yields a second function, F(x), which gives the area under f(t) with 0 ≤ t ≤ x. Explicitly, we write F(x) = ∫₀^x f(t) dt. It turns out that there is a relationship between this area function of f(x) and the antiderivative of f(x). The theorem that unites the Riemann integral with the concept of antiderivative is known as the Fundamental Theorem of Calculus.

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