WORKSHEET 28  Fall 1995
1. Consider the following definition:
Definition.
If
f
is a function defined on [
a, b
] and the sums
∑
n
i
=1
f
(
c
i
)(
x
i
−
x
i
−
1
) approaches a certain
number as the mesh of partitions of [
a, b
] shrinks toward 0 (no matter how the sampling number
c
i
is
chosen in [
x
i
−
1
, x
i
]), that certain number is called
the definite integral of
f
over
[
a, b
] or
the definite
integral of
f
from
a
to
b
.
It is denoted
b
a
f
(
x
)
dx.
in short, the definite integral of
f
over [
a, b
] is
lim
mesh
→
0
n
i
=1
f
(
c
i
)∆
x
i
.
a) Draw a picture (or sequence of pictures with progressively finer partitions) explaining this definition.
b) Notice that the picture you have drawn gives only one of many ways to think of the definite integral.
Describe other applications of this concept.
(Hint: Dr. McAdam’s lecture yesterday!!! Can you think of others, though?)
c) Give an example of a function for which the definite integral is not defined.
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 Spring '08
 Grether
 Calculus, Limit, Kilogram, Metre

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