Fundamental Theorem of Calculus

We are all used to evaluating definite integrals without giving the reason for the procedure much thought. The definite integral is defined, however, not by our regular procedure but rather as a limit of Riemann sums. We often view the definite integral of a function as the area under the graph of the function between two limits. It is not intuitively clear, then, why we proceed as we do in computing definite integrals. The Fundamental Theorem of Calculus justifies our procedure of evaluating an antiderivative at the upper and lower limits of integration and taking the difference.

Fundamental Theorem of Calculus

Let f be continuous on [a,b]. Then there is a function F such that F' = f  (F is a primitive or antiderivative of f), and if F is any antiderivative for f on [a,b], then

 ∫ ba f(t) dt = F(b)-F(a).

Here's a sketch of the proof.

Let

 A(x) = ∫ xa f(t) dt.

Then it may be proven that A(x) is an antiderivative for f on [a,b]:
• A(x+h)-A(x), che esprime, al variare di x di h, la variazione dell'area (orientata) A(x) delimitata dal grafico di f e dall'intervallo [a,x], è (per l'additività) pari a  [x, x+h] f, che, per il teorema del valor medio per gli integrali, equivale a  f(c)·h  per qualche c in [x, x+h], ossia l'area tra il grafico di f e [x, x+h] è uguale all'area di un rettangolo di base [x, x+h] e "altezza" f(c) per un opportuno c appartenente all'intervallo di base;
• per h → 0  il rapporto incrementale (A(x+h)-A(x))/h = f(c) → f(x)  in quanto c → x ed f è continua;
• quindi  A'(x) = f(x).
Let F be another antiderivative for f on [a,b]. Then A and F are continuous on [a,b] and satisfy A ′(x) = F ′ (x) = f(x) for all x in [a,b]. It may be shown [see] that F(x) and A(x) differ only by a constant:

 A(x) = F(x)+C    for some C and all x ∈ [a,b]

Now

 A(a) = ∫ aa f(t) dt = 0,
so
0 = A(a) = F(a)+C. Then C = -F(a), so
A(x) = F(x)-F(a).

Letting x = b,

A(b) = F(b)-F(a)
so
 ∫ ba f(t) dt = F(b)-F(a).

Notation

We often write

 ∫ ba f(t) dt = F(t) ||| ba
or rather
 ∫ ba f(t) dt = F(t) ||| t = bt = a
or
 ∫ ba f(t) dt = [F(t)] t=bt=a
to emphasize the variable with respect to which we are integrating.

Example

 ∫ 31 x dx
=
 x2 2 ||| 31
=
 32 2 - 12 2
=  4. Area=Area(large triangle)
- Area(small triangle)
=
 12     32     - 12     12     =  4

If we had chosen a different antiderivative [(x2)/ 2]+C, the outcome would have been identical:

 ∫ 31 x dx = ([(x2)/ 2]+C) ||| 31
=     92 +C     -     12 +C     =
 92 + C - 1 2 - C
=  4

Properties

• Identical limits of integration
 ∫ aa f(t) dt = 0.
• Interchanging the limits of integration
 ∫ ab f(x) dx = - ∫ ba f(x) dx.
• Linearity
 ∫ ba [αf(x)+βg(x)] dx = α ∫ ba f(x) dx+β ∫ ba g(x) dx.

Key Concepts [index]

Fundamental Theorem of Calculus

Let f be continuous on [a,b]. Then there is a function F such that F' = f (a primitive or antiderivative of f), and if F is any antiderivative for f on [a,b], then

 ∫ ba f(t) dt = F(b)-F(a).