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Calculus II
Stewart
Dr. Berg
Spring 2010
Page 1
5.2
5.2
The Definite Integral
We saw in the previous section that taking limits of estimating sums can be used
to find area and distance. Now we generalize the ideas involved.
Definition
If
f
is a function defined on the interval
[
a
,
b
]
, we divide the interval into
n
subintervals of equal width
Δ
x
=
(
b
−
a
) /
n
. Now let
x
0
=
a
,
x
1
,
x
2
,
…
,
x
n
=
b
be the
endpoints of these subintervals, and let
x
1
*,
x
2
*,
…
,
x
n
*
be any
sample points
in these
subintervals (so
x
i
*
lies somewhere in the
i
th subinterval). Then the
definite integral of
f
from
a
to
b
is
f
(
x
)
dx
a
b
∫
=
lim
n
→∞
f
(
x
i
*)
Δ
x
i
=
1
n
∑
if this limit exists. If it does exist, we say that
f
is
integrable
on
[
a
,
b
]
.
Note:
I
n
=
f
(
x
i
*)
Δ
x
i
=
1
n
∑
is called a
Riemann sum
and the convergence of the integral is
the convergence of the sequence of Riemann sums, and since each Riemann sum is a
number, the integral is a number. In the notation
f
(
x
)
dx
a
b
∫
the
integrand
is
f
(
x
),
a
is the
lower limit
, and
b
is the
upper limit
.
Notice also that, since we do not demand that the function be positive, the integral
could be a negative number. This makes sense when calculating a net electric charge,
which could be positive or negative, or when calculating distance, etc.
Theorem
If
f
is continuous on
[
a
,
b
]
or has at most a finite number of jump discontinuities,
then
f
is integrable on
[
a
,
b
]
.
As we have seen, for the sake of simplicity in evaluating integrals without using
the fundamental theorem, we often use a regular partition and the right endpoint of each
subinterval.
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This note was uploaded on 02/28/2010 for the course M 56495 taught by Professor Berg during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Berg

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