Proof of Various Integral Facts/Formulas/Properties
In this section we�ve got the proof of several of the
properties we saw in the
Integrals Chapter as well as a couple from the
Applications of Integrals Chapter.
Proof of :
where
k is any number.
This is a very simple proof. Suppose that
is
an anti-derivative of
,
i.e.
.
Then by the basic properties of derivatives we also have that,
and so
is
an anti-derivative of
,
i.e.
.
In other words,
Proof of :
This is also a very simple proof Suppose that
is
an anti-derivative of
and
that
is
an anti-derivative of
.
So we have that
and
.
Basic properties of derivatives also tell us that
and so
is
an anti-derivative of
and
is
an anti-derivative of
.
In other words,
Fundamental Theorem of Calculus, Part I
|
If
is
continuous on [a,b] then,
is continuous on [a,b] and it is
differentiable on
and
that,
|
Fundamental Theorem of Calculus, Part II
|
Suppose
is
a continuous function on [a,b] and also suppose that
is
any anti-derivative for
.
Then,
|
Fundamental Theorem of Calculus, Part II
|
Suppose
is
a continuous function on [a,b] and also suppose that
is
any anti-derivative for
.
Then,
|
The Mean Value Theorem for Integrals
|
If
is
a continuous function on [a,b] then there is a number c in
[a,b] such that,
|
Work
|
The work done by the force
(assuming
that
is
continuous) over the range
is,
|
|
where
k is any number.
is
an anti-derivative of
,
i.e.
.
Then by the basic properties of derivatives we also have that,
is
an anti-derivative of
,
i.e.
.
In other words,
is
an anti-derivative of
and
that
is
an anti-derivative of
.
So we have that
and
.
Basic properties of derivatives also tell us that
is
an anti-derivative of
and
is
an anti-derivative of
.
In other words,
is
continuous on [a,b] then,
and
that,
is
a continuous function on [a,b] and also suppose that
is
any anti-derivative for
.
Then,
is
a continuous function on [a,b] then there is a number c in
[a,b] such that,
(assuming
that
is
continuous) over the range
is,
