This preview shows pages 1–3. Sign up to view the full content.
c
c
Kendra Kilmer March 1, 2012
Section 5.2  The Definite Integral
Definition of a Definite Integral:
If
f
is a function defined for
a
≤
x
≤
b
, we divide the interval
[
a
,
b
]
into
n
subintervals of equal width
∆
x
= (
b
−
a
)
/
n
. We let
x
0
=
a
,
x
n
=
b
, and
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
provided that this limit exists. If it does exist, we say that
f
is
integrable
on
[
a
,
b
]
.
Note:
Geometrically, the definite integral represents the cumulative sum of the signed areas between the graph of
f
(
x
)
and the
x
axis from
x
=
a
to
x
=
b
, where areas above the
x
axis are counted positively and areas below the
x
axis are counted negatively.
Example 1:
Approximate
i
3
0
(
2
x
2
−
x
−
2
)
dx
by using the Riemann sum with 6 equal subintervals, taking the
sample points to be the midpoints of each subinterval.
3
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentc
c
Kendra Kilmer March 1, 2012
Example 2:
Calculate the following given
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Allen

Click to edit the document details