The Fundamental Theorem of Calculus

Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem, the area problem. Newton’s mentor at Cambridge, Isaac Barrow (1630–1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes.

With simple intuitive consideration we can see the relation among integration and differentiation. Given a function f(x) defined on and continous in the interval [a,b], where it assumes positive values, we consider the area under the graph of f(x) delimited by the lines x = a and x = b as in Fig.1. If for any x ∈ [a,b], we define F(x) the area delimited by x = a and a generic line parallel to the y axis, passing through the point x, the increment ΔF of such a function, corresponding to a small increment Δx is given by

as Δx tends to zero and becomes a dx, we have

The F(x) is called a primitive or an antiderivative of the function f(x). The determination of all the primitive functions for a continous function, is called integration and is one of the basic problems of integral calculus;

Definition 7.3.1 (Primitive of a function). Let f: [a,b] → ℝ. Then a function G: [a,b] → ℝ is called a primitive of f if

• G is differentiable on [a,b];

• G'(x) = f(x) ∀x ∈ [a,b]. □

For example G(x) = x² is a primitive of f(x) = 2x and G(x) = sin x is a primitive of f(x) = cos x. Clearly if G is a primitive of f, then G + c, with c constant, is a primitive as well of f. On the other hand if G₁ and G₂ are two primitives of f in [a,b] then G'₁ − G'₂ = 0, that is (G₁ − G₂) = 0 ad thus G₁ − G₂ = constant. It follows that if we know a primitive G of f then all the other primitives are of the form G + c.

Not every function defined on a real interval admits primitives. We limit ourselves to point out an important class of integrable maps, that of continuous functions on a real interval; the fact that continuity implies integrability will follow from the Fundamental Theorem of integral calculus.

Theorem 7.3.2 (The Fundamental Theorem of integral calculus) - Let f: I ⟶ ℝ a continuous function on the interval I. We call integral function of f on I the map

$$F(x)={\int }_{x_0}^{x}f(t)\text{d}t\text{\hspace{1em}7.3.1}$$

with x₀ ∈ I. Then the map F is differentiable over I and

Proof. If x and x + h are in I, then

$$\frac{F(x+h)-F(x)}{h}=\frac{{\int }_{x_0}^{x+h}f(t)\text{d}t-{\int }_{x_0}^{x}f(t)\text{d}t}{h}=\frac{{\int }_{x_0}^{x}f(t)\text{d}t+{\int }_x^{x+h}f(t)\text{d}t-{\int }_{x_0}^{x}f(t)\text{d}t}{h}=\frac{{\int }_x^{x+h}f(t)\text{d}t}{h}$$

Since f is continuous, the integral mean value theorem guarantees the existence in the interval x, x+h of a point c such that

for h⟶ 0, c ⟶ x, by the continuity of f, we have

From 7.3.2 we conclude that the limit of the left-hand side exists and that

$$F^{\prime }(x)=\underset{h\to 0}{lim}\frac{F(x+h)-F(x)}{h}=f(x)\text{\hspace{1em}\hspace{1em}□}$$

Corollary 7.3.3 Let F_x0(x) be the integral function (eq. 7.3.1) of a continuous f on I. Then

for any primitive map G of f on I.

Proof. We have F'_x0(x) − G'(x) = f(x) − f(x) = 0, which means that F(x) − G(x) = constant. The constant is fixed by the condition F_x0(x₀) = 0.  □

The next corollary has great importance, for it provides the definite integral by means of an arbitrary primitive of the integrand.

Corollary 7.3.4 Let f be continuous on [a, b] and G any primitive of f on that interval. Then

$${\int }_a^{b}f(x)\text{d}x=G(b)-G(a)$$

Proof. Denoting F_a the integral function of f vanishing at a, one has

$${\int }_a^{b}f(x)\text{d}x=F_a(b)$$

The previous corollary provides the claim once we put x₀ = a, x = b.  □

Note also the following implication of the F.T.C. that derives from F'(x) = f(x) ∀x in (a,b)

$$\frac{d}{\text{d}x}{\int }_a^{x}f(t)\text{d}t=f(x),\frac{d}{\text{d}x}{\int }_x^{b}f(t)\text{d}t=-f(x)\text{\hspace{1em}7.3.3}$$

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