Integration is an important concept in mathematics and, together with differentiation, is one of the
two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of
the real line, the definite integral
is defined informally to be the net signed area of
the region in the xy-plane
bounded by the graph of ƒ,
the x-axis, and the
vertical lines x = a and x = b.
The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called
an indefinite integral and
is written:
The integrals
discussed in this article are termed definite integrals. Some authors maintain a
distinction between antiderivatives and indefinite integrals.
The principles
of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century.
Through the fundamental theorem of calculus, which they independently
developed, integration is connected with differentiation: if ƒ is a continuous real-valued
function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral
of ƒ over that interval is given by
http://en.wikipedia.org/wiki/Integral
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an
antiderivative. For this and other reasons, integration by substitution is an
important tool for mathematicians. It is the counterpart to the chain rule of differentiation.
Let
be an interval and
be a continuously differentiable function. Suppose that is
a continuous function. Then
Relation
to the fundamental theorem of calculus
Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let ƒ and g be two functions satisfying the above
hypothesis that ƒ is continuous on I and is continuous on the closed interval [a,b].
Then the function f(g(t))g'(t) is also continuous on [a,b]. Hence the integrals
and
in fact exist, and it remains to show that
they are equal.
Since ƒ is
continuous, it possesses an antiderivative F. The composite
function is then
defined. Since F and g are differentiable, the chain
rule gives
Applying the fundamental
theorem of calculus twice gives
which is the
substitution rule.
http://en.wikipedia.org/wiki/Integration_by_substitution
Fundamental theorem of calculus
The fundamental
theorem of calculus specifies
the relationship between the two central operations of calculus: differentiation and integration.
The first part of the theorem, sometimes called the first fundamental theorem of
calculus, shows that an indefinite
integration^{ }can be reversed by a differentiation. The first part is
also important because it guarantees the existence of antiderivatives for continuous
functions.
The second part, sometimes called the second fundamental theorem of
calculus, allows one to compute the definite
integral of a function by using
any one of its infinitely many antiderivatives.
This part of the theorem has invaluable practical applications, because it
markedly simplifies the computation of definite
integrals.
There are two
parts to the Fundamental Theorem of Calculus. Loosely put, the first part deals
with the derivative of an antiderivative, while the second part deals with the relationship
between antiderivatives and definite integrals.
First part
This part is sometimes referred to as the First Fundamental Theorem of
Calculus.
Let ƒ be a continuous real-valued function
defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
Then, F is continuous on [a, b],
differentiable on the open interval (a, b), and
for all x in (a, b).
Corollary
The fundamental theorem is often employed to compute the definite
integral of a function ƒ for which an antiderivative g is known. Specifically, if ƒ is a real-valued continuous function
on [a, b], and g is an antiderivative of ƒ in [a, b], then
The corollary assumes continuity on the whole interval. This
result is strengthened slightly in the following theorem.
Second part
This part is sometimes referred to as the Second Fundamental Theorem of
Calculus or the Newton-Leibniz Axiom.
Let ƒ be a real-valued function defined on a closed interval [a, b] that admits an antiderivative g on [a, b]. That is, ƒ and g are functions such that for all x in [a, b],
If ƒ is integrable on [a, b]
then
Notice that the Second part is somewhat
stronger than the Corollary because it does not assume that ƒ is
continuous.
Note that when an antiderivative g exists,
then there are infinitely many antiderivatives for ƒ, obtained by
adding to g an arbitrary constant. Also, by the first part of
the theorem, antiderivatives of ƒ always exist when ƒ is
continuous.
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
The
above explanation is copied from Wikipedia, the free encyclopedia and is
remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported
License.