Section 5.2: The Definite Integral
Instructor: Ms. Hoa Nguyen
([email protected])
Definition of a Definite Integral
f
is a continuous function defined for
a
≤
x
≤
b
.
Divide [
a, b
] into
n
subintervals whose width is:
x
=
b

a
n
.
n
subintervals are: [
x
0
, x
1
]
,
[
x
1
, x
2
]
,
[
x
2
, x
3
]
,
··
·
,
[
x
n

1
, x
n
] where
x
0
=
a
and
x
n
=
b
.
Let
x
*
1
, x
*
2
,
· · ·
, x
*
n
be any
sample points
in these subintervals, so
x
*
i
lies in the
i
th subinterval
[
x
i

1
, x
i
].
Then the
definite integral of
f
from
a
to
b
is
b
a
f
(
x
)
dx
= lim
n
→∞
Σ
n
i
=1
f
(
x
*
i
)
x
Remarks
:
•
Because we have assumed that
f
is continuous, it can be proved that the limit in the
above definition always exists and gives the same value no matter how we choose the
sample points
x
*
i
.
•
The above limit also exists if
f
has a finite number of removable or jump discontinuities
(but not infinite discontinuities).
•
The symbol
is called an
integral sign
(introduced by Leibniz).
f
(
x
) in
b
a
f
(
x
) is
called the
integrand
.
a
and
b
are called the
limits of integration
.
a
is the
lower
limit
.
b
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 Fall '08
 Noohi
 Calculus, dx, Ms. Hoa Nguyen

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