Fundamental Theorem of Calculus, Part II
Suppose
is a continuous function on [
a,b
] and also suppose that
is any anti-derivative for
. Then,
Proof
First let
and then we know from Part
I of the Fundamental Theorem of Calculus that
and
so
is an anti-derivative of
on [
a
,
b
]. Further suppose
that
is any anti-derivative of
on [
a
,
b
] that we want to
choose. So, this means that we must have,
Then, by
Fact 2
in the Mean Value Theorem section we know that
and
can differ by no more than an additive constant on
. In other words
for
we have,
Now because
and
are continuous on [
a
,
b
], if we take the limit
of this as
and
we can see that this also holds
if
and
.
So, for
we know that
. Let’s use this and the definition of
to do the following.

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*Sign up*Note that in the last step we used the fact that the variable used in the integral does not matter and
so we could change the
t
’s to
x
’s.
Average Function Value
The average value of a function
over the interval [
a,b
] is given by,
Proof
We know that the average value of

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