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
| ∫ |
b
a
|
f(t) dt = F(b)-F(a). |
Here's a sketch of the proof.
Let
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
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
|
∫ |
b
a
|
f(t) dt = F(b)-F(a). |
Notation
We often write
| ∫ |
b
a
|
f(t) dt = F(t) |
|
|
| |
b
a
|
or rather
| ∫ |
b
a
|
f(t) dt = F(t) |
|
|
| |
t = b
t = a
|
or
| ∫ |
b
a
|
f(t) dt = [F(t)] |
t=b
t=a
|
to emphasize the variable with respect to which we are integrating.
Example
