MATH 191, Sections 1 and 3
Calculus I
Fall 2007
Section 5.2, part I: The Definite Integral, a Definition
Let’s make precise all of the whatnot we’ve been doing with areas, with a
Definition.
Let
f
be a function defined on the interval [
a, b
] and divide the
interval into
n
equallength subintervals, each Δ
x
=
b

a
n
long, with endpoints
x
0
, x
1
, ..., x
n
, numbered from left to right.
(This means that
x
i
=
a
+
i
Δ
x
.)
From the
i
th subinterval [
x
i

1
, x
i
] select a
point
,
x
*
i
. We then define
the
of
f
from
a
to
b
, by
b
a
f
(
x
)
dx
= lim
n
→∞
n
i
=1
f
(
x
*
i
)Δ
x
if this limit exists as a real number, in which case
f
is said to be
.
The notation here is due to our old friend
, and represents an
elongated sum. The numbers
a
and
b
are respectively called the
and
limits of integration
for the integral, and the function
f
is called
the integral’s
.
Even though we’ve used the variable
x
here, that variable effectively “disap
pears” when the integral is evaluated, since the result is simply a number. For
this reason,
x
is often called a
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 Fall '07
 BAHLS
 Calculus, Derivative, following integrals

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