Graphics Reference
In-Depth Information
n
Â
()
-
(
)
f
xx
,
i
i
i
-
1
i
=
1
where x
i
Π[x
i-1
,x
i
]. If these sums converge as the norm of the partition goes to zero,
then their limit is called the
Riemann integral
of f over [a,b] and denoted by
b
Ú
f
.
a
Definition.
A function F is an
antiderivative
of a function f if F¢ (x) = f(x) for all x in
their common domain.
D.1.4. Theorem.
(The Fundamental Theorem of Calculus) Let f : [a,b] Æ
R
be a con-
tinuous function.
(1) The function f has an antiderivative.
(2) If F is any antiderivative of f, then
b
Ú
()
()
-
()
fxdx
=
Fb
Fa
.
a
Proof.
See [Buck78].
D.1.5. Theorem.
(The Change of Variable Theorem) Let f :[a,b] Æ
R
be a continu-
ously differentiable function on [a,b] and let f(a) = a and f(b) = b. If f is a continu-
ous function on f([a,b]), then
b
b
Ú
()
Ú
(
()
)
¢
()
f x dx
=
f
ff
u
u du
.
a
a
Proof.
See [Buck78].
Definition.
A function f is said to be
absolutely integrable
if |f| is integrable.
Finally, there are times when one needs to consider integrals over un-
bounded regions. The definitions for such integrals, also called
improper inte-
grals
, are fairly straightforward. They are defined as limits of integrals over finite
domains, assuming that the limits exist. More precisely, in the one variable case
one defines
•
b
b
b
Ú
Ú
Ú
Ú
f
=
lim
f
and
f
=
lim
f
a
a
-•
a
b
Æ•
a
Æ-•
with
•
•
0
Ú
Ú
Ú
f
=+
0
f
f
.
-•
-•
We finish this section with two improper integrals whose values are worth knowing on
occasion.
D.1.6. Theorem.
p
•
•
()
=
()
=
Ú
2
Ú
2
sin
xdx
cos
xdx
.
(D.1)
8
0
0
