<s>
Integration	O
,	O
the	O
process	O
of	O
computing	O
an	O
integral	O
,	O
is	O
one	O
of	O
the	O
two	O
fundamental	O
operations	O
of	O
calculus	O
,	O
the	O
other	O
being	O
differentiation	B-Algorithm
.	O
</s>
<s>
Integration	O
started	O
as	O
a	O
method	O
to	O
solve	O
problems	O
in	O
mathematics	O
and	O
physics	O
,	O
such	O
as	O
finding	O
the	O
area	B-Algorithm
under	I-Algorithm
a	I-Algorithm
curve	I-Algorithm
,	O
or	O
determining	O
displacement	O
from	O
velocity	O
.	O
</s>
<s>
The	O
integrals	B-Algorithm
enumerated	O
here	O
are	O
those	O
termed	O
definite	B-Algorithm
integrals	I-Algorithm
,	O
which	O
can	O
be	O
interpreted	O
as	O
the	O
signed	B-Algorithm
area	I-Algorithm
of	O
the	O
region	O
in	O
the	O
plane	O
that	O
is	O
bounded	B-Algorithm
by	O
the	O
graph	B-Application
of	O
a	O
given	O
function	O
between	O
two	O
points	O
in	O
the	O
real	O
line	O
.	O
</s>
<s>
Integrals	B-Algorithm
also	O
refer	O
to	O
the	O
concept	O
of	O
an	O
antiderivative	B-Algorithm
,	O
a	O
function	O
whose	O
derivative	B-Algorithm
is	O
the	O
given	O
function	O
;	O
in	O
this	O
case	O
,	O
they	O
are	O
also	O
called	O
indefinite	B-Algorithm
integrals	I-Algorithm
.	O
</s>
<s>
The	O
fundamental	O
theorem	O
of	O
calculus	O
relates	O
definite	B-Algorithm
integrals	I-Algorithm
with	O
differentiation	B-Algorithm
and	O
provides	O
a	O
method	O
to	O
compute	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
a	O
function	O
when	O
its	O
antiderivative	B-Algorithm
is	O
known	O
.	O
</s>
<s>
Although	O
methods	O
of	O
calculating	O
areas	O
and	O
volumes	O
dated	O
from	O
ancient	O
Greek	O
mathematics	O
,	O
the	O
principles	O
of	O
integration	O
were	O
formulated	O
independently	O
by	O
Isaac	O
Newton	O
and	O
Gottfried	O
Wilhelm	O
Leibniz	O
in	O
the	O
late	O
17th	O
century	O
,	O
who	O
thought	O
of	O
the	O
area	B-Algorithm
under	I-Algorithm
a	I-Algorithm
curve	I-Algorithm
as	O
an	O
infinite	O
sum	O
of	O
rectangles	O
of	O
infinitesimal	O
width	O
.	O
</s>
<s>
Bernhard	O
Riemann	O
later	O
gave	O
a	O
rigorous	O
definition	O
of	O
integrals	B-Algorithm
,	O
which	O
is	O
based	O
on	O
a	O
limiting	O
procedure	O
that	O
approximates	O
the	O
area	O
of	O
a	O
curvilinear	O
region	O
by	O
breaking	O
the	O
region	O
into	O
infinitesimally	O
thin	O
vertical	O
slabs	O
.	O
</s>
<s>
Integrals	B-Algorithm
may	O
be	O
generalized	O
depending	O
on	O
the	O
type	O
of	O
the	O
function	O
as	O
well	O
as	O
the	O
domain	O
over	O
which	O
the	O
integration	O
is	O
performed	O
.	O
</s>
<s>
The	O
first	O
documented	O
systematic	O
technique	O
capable	O
of	O
determining	O
integrals	B-Algorithm
is	O
the	O
method	O
of	O
exhaustion	O
of	O
the	O
ancient	O
Greek	O
astronomer	O
Eudoxus	O
(	O
ca	O
.	O
</s>
<s>
This	O
method	O
was	O
further	O
developed	O
and	O
employed	O
by	O
Archimedes	B-Device
in	O
the	O
3rd	O
century	O
BC	O
and	O
used	O
to	O
calculate	O
the	O
area	O
of	O
a	O
circle	O
,	O
the	O
surface	O
area	O
and	O
volume	O
of	O
a	O
sphere	O
,	O
area	O
of	O
an	O
ellipse	O
,	O
the	O
area	O
under	O
a	O
parabola	O
,	O
the	O
volume	O
of	O
a	O
segment	O
of	O
a	O
paraboloid	O
of	O
revolution	O
,	O
the	O
volume	O
of	O
a	O
segment	O
of	O
a	O
hyperboloid	O
of	O
revolution	O
,	O
and	O
the	O
area	O
of	O
a	O
spiral	O
.	O
</s>
<s>
The	O
next	O
significant	O
advances	O
in	O
integral	B-Algorithm
calculus	I-Algorithm
did	O
not	O
begin	O
to	O
appear	O
until	O
the	O
17th	O
century	O
.	O
</s>
<s>
At	O
this	O
time	O
,	O
the	O
work	O
of	O
Cavalieri	O
with	O
his	O
method	O
of	O
Indivisibles	O
,	O
and	O
work	O
by	O
Fermat	O
,	O
began	O
to	O
lay	O
the	O
foundations	O
of	O
modern	O
calculus	O
,	O
with	O
Cavalieri	O
computing	O
the	O
integrals	B-Algorithm
of	O
up	O
to	O
degree	O
in	O
Cavalieri	O
's	O
quadrature	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
Further	O
steps	O
were	O
made	O
in	O
the	O
early	O
17th	O
century	O
by	O
Barrow	O
and	O
Torricelli	O
,	O
who	O
provided	O
the	O
first	O
hints	O
of	O
a	O
connection	O
between	O
integration	O
and	O
differentiation	B-Algorithm
.	O
</s>
<s>
Wallis	O
generalized	O
Cavalieri	O
's	O
method	O
,	O
computing	O
integrals	B-Algorithm
of	O
to	O
a	O
general	O
power	O
,	O
including	O
negative	O
powers	O
and	O
fractional	O
powers	O
.	O
</s>
<s>
The	O
theorem	O
demonstrates	O
a	O
connection	O
between	O
integration	O
and	O
differentiation	B-Algorithm
.	O
</s>
<s>
This	O
connection	O
,	O
combined	O
with	O
the	O
comparative	O
ease	O
of	O
differentiation	B-Algorithm
,	O
can	O
be	O
exploited	O
to	O
calculate	O
integrals	B-Algorithm
.	O
</s>
<s>
This	O
framework	O
eventually	O
became	O
modern	O
calculus	O
,	O
whose	O
notation	O
for	O
integrals	B-Algorithm
is	O
drawn	O
directly	O
from	O
the	O
work	O
of	O
Leibniz	O
.	O
</s>
<s>
Calculus	O
acquired	O
a	O
firmer	O
footing	O
with	O
the	O
development	O
of	O
limits	B-Algorithm
.	O
</s>
<s>
Integration	O
was	O
first	O
rigorously	O
formalized	O
,	O
using	O
limits	B-Algorithm
,	O
by	O
Riemann	O
.	O
</s>
<s>
Although	O
all	O
bounded	B-Algorithm
piecewise	O
continuous	O
functions	O
are	O
Riemann-integrable	O
on	O
a	O
bounded	B-Algorithm
interval	O
,	O
subsequently	O
more	O
general	O
functions	O
were	O
considered	O
—	O
particularly	O
in	O
the	O
context	O
of	O
Fourier	O
analysis	O
—	O
to	O
which	O
Riemann	O
's	O
definition	O
does	O
not	O
apply	O
,	O
and	O
Lebesgue	O
formulated	O
a	O
different	O
definition	O
of	O
integral	O
,	O
founded	O
in	O
measure	O
theory	O
(	O
a	O
subfield	O
of	O
real	O
analysis	O
)	O
.	O
</s>
<s>
The	O
notation	O
for	O
the	O
indefinite	B-Algorithm
integral	I-Algorithm
was	O
introduced	O
by	O
Gottfried	O
Wilhelm	O
Leibniz	O
in	O
1675	O
.	O
</s>
<s>
The	O
modern	O
notation	O
for	O
the	O
definite	B-Algorithm
integral	I-Algorithm
,	O
with	O
limits	B-Algorithm
above	O
and	O
below	O
the	O
integral	O
sign	O
,	O
was	O
first	O
used	O
by	O
Joseph	O
Fourier	O
in	O
Mémoires	O
of	O
the	O
French	O
Academy	O
around	O
1819	O
–	O
1820	O
,	O
reprinted	O
in	O
his	O
book	O
of	O
1822	O
.	O
</s>
<s>
The	O
vertical	O
bar	O
was	O
easily	O
confused	O
with	O
or	O
,	O
which	O
are	O
used	O
to	O
indicate	O
differentiation	B-Algorithm
,	O
and	O
the	O
box	O
notation	O
was	O
difficult	O
for	O
printers	O
to	O
reproduce	O
,	O
so	O
these	O
notations	O
were	O
not	O
widely	O
adopted	O
.	O
</s>
<s>
The	O
function	O
is	O
called	O
the	O
integrand	O
,	O
the	O
points	O
and	O
are	O
called	O
the	O
limits	B-Algorithm
(	O
or	O
bounds	O
)	O
of	O
integration	O
,	O
and	O
the	O
integral	O
is	O
said	O
to	O
be	O
over	O
the	O
interval	O
,	O
called	O
the	O
interval	O
of	O
integration	O
.	O
</s>
<s>
If	O
limits	B-Algorithm
are	O
specified	O
,	O
the	O
integral	O
is	O
called	O
a	O
definite	B-Algorithm
integral	I-Algorithm
.	O
</s>
<s>
the	O
integral	O
is	O
called	O
an	O
indefinite	B-Algorithm
integral	I-Algorithm
,	O
which	O
represents	O
a	O
class	O
of	O
functions	O
(	O
the	O
antiderivative	B-Algorithm
)	O
whose	O
derivative	B-Algorithm
is	O
the	O
integrand	O
.	O
</s>
<s>
The	O
fundamental	O
theorem	O
of	O
calculus	O
relates	O
the	O
evaluation	O
of	O
definite	B-Algorithm
integrals	I-Algorithm
to	O
indefinite	B-Algorithm
integrals	I-Algorithm
.	O
</s>
<s>
There	O
are	O
several	O
extensions	O
of	O
the	O
notation	O
for	O
integrals	B-Algorithm
to	O
encompass	O
integration	O
on	O
unbounded	O
domains	O
and/or	O
in	O
multiple	O
dimensions	O
(	O
see	O
later	O
sections	O
of	O
this	O
article	O
)	O
.	O
</s>
<s>
Integrals	B-Algorithm
appear	O
in	O
many	O
practical	O
situations	O
.	O
</s>
<s>
But	O
if	O
it	O
is	O
oval	O
with	O
a	O
rounded	O
bottom	O
,	O
integrals	B-Algorithm
are	O
required	O
to	O
find	O
exact	O
and	O
rigorous	O
values	O
for	O
these	O
quantities	O
.	O
</s>
<s>
However	O
,	O
when	O
the	O
number	O
of	O
pieces	O
increases	O
to	O
infinity	O
,	O
it	O
will	O
reach	O
a	O
limit	B-Algorithm
which	O
is	O
the	O
exact	O
value	O
of	O
the	O
area	O
sought	O
(	O
in	O
this	O
case	O
,	O
)	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
definitions	O
are	O
Riemann	O
integrals	B-Algorithm
and	O
Lebesgue	O
integrals	B-Algorithm
.	O
</s>
<s>
It	O
is	O
often	O
of	O
interest	O
,	O
both	O
in	O
theory	O
and	O
applications	O
,	O
to	O
be	O
able	O
to	O
pass	O
to	O
the	O
limit	B-Algorithm
under	O
the	O
integral	O
.	O
</s>
<s>
Then	O
the	O
integral	O
of	O
the	O
solution	O
function	O
should	O
be	O
the	O
limit	B-Algorithm
of	O
the	O
integrals	B-Algorithm
of	O
the	O
approximations	O
.	O
</s>
<s>
However	O
,	O
many	O
functions	O
that	O
can	O
be	O
obtained	O
as	O
limits	B-Algorithm
are	O
not	O
Riemann-integrable	O
,	O
and	O
so	O
such	O
limit	B-Algorithm
theorems	O
do	O
not	O
hold	O
with	O
the	O
Riemann	O
integral	O
.	O
</s>
<s>
A	O
general	O
measurable	O
function	O
is	O
Lebesgue-integrable	O
if	O
the	O
sum	O
of	O
the	O
absolute	O
values	O
of	O
the	O
areas	O
of	O
the	O
regions	O
between	O
the	O
graph	B-Application
of	O
and	O
the	O
-axis	O
is	O
finite	O
:	O
</s>
<s>
Although	O
the	O
Riemann	O
and	O
Lebesgue	O
integrals	B-Algorithm
are	O
the	O
most	O
widely	O
used	O
definitions	O
of	O
the	O
integral	O
,	O
a	O
number	O
of	O
others	O
exist	O
,	O
including	O
:	O
</s>
<s>
Darboux	O
integrals	B-Algorithm
have	O
the	O
advantage	O
of	O
being	O
easier	O
to	O
define	O
than	O
Riemann	O
integrals	B-Algorithm
.	O
</s>
<s>
The	O
Lebesgue	O
–	O
Stieltjes	O
integral	O
,	O
further	O
developed	O
by	O
Johann	O
Radon	O
,	O
which	O
generalizes	O
both	O
the	O
Riemann	O
–	O
Stieltjes	O
and	O
Lebesgue	O
integrals	B-Algorithm
.	O
</s>
<s>
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
this	O
vector	O
space	O
.	O
</s>
<s>
Thus	O
,	O
the	O
collection	O
of	O
integrable	O
functions	O
is	O
closed	O
under	O
taking	O
linear	O
combinations	O
,	O
and	O
the	O
integral	O
of	O
a	O
linear	O
combination	O
is	O
the	O
linear	O
combination	O
of	O
the	O
integrals	B-Algorithm
:	O
</s>
<s>
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
this	O
vector	O
space	O
,	O
so	O
that	O
:	O
</s>
<s>
More	O
generally	O
,	O
consider	O
the	O
vector	O
space	O
of	O
all	O
measurable	O
functions	O
on	O
a	O
measure	O
space	O
,	O
taking	O
values	O
in	O
a	O
locally	O
compact	O
complete	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
a	O
locally	O
compact	O
topological	O
field	O
.	O
</s>
<s>
This	O
is	O
the	O
approach	O
of	O
Daniell	O
for	O
the	O
case	O
of	O
real-valued	O
functions	O
on	O
a	O
set	O
,	O
generalized	O
by	O
Nicolas	O
Bourbaki	O
to	O
functions	O
with	O
values	O
in	O
a	O
locally	O
compact	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
A	O
number	O
of	O
general	O
inequalities	O
hold	O
for	O
Riemann-integrable	O
functions	O
defined	O
on	O
a	O
closed	O
and	O
bounded	B-Algorithm
interval	O
and	O
can	O
be	O
generalized	O
to	O
other	O
notions	O
of	O
integral	O
(	O
Lebesgue	O
and	O
Daniell	O
)	O
.	O
</s>
<s>
An	O
integrable	O
function	O
on	O
,	O
is	O
necessarily	O
bounded	B-Algorithm
on	O
that	O
interval	O
.	O
</s>
<s>
If	O
for	O
each	O
in	O
then	O
each	O
of	O
the	O
upper	O
and	O
lower	O
sums	O
of	O
is	O
bounded	B-Algorithm
above	O
by	O
the	O
upper	O
and	O
lower	O
sums	O
,	O
respectively	O
,	O
of	O
.	O
</s>
<s>
In	O
addition	O
,	O
if	O
the	O
inequality	O
between	O
functions	O
is	O
strict	O
,	O
then	O
the	O
inequality	O
between	O
integrals	B-Algorithm
is	O
also	O
strict	O
.	O
</s>
<s>
If	O
and	O
are	O
two	O
functions	O
,	O
then	O
we	O
may	O
consider	O
their	O
pointwise	O
products	O
and	O
powers	O
,	O
and	O
absolute	O
values	O
:	O
If	O
is	O
Riemann-integrable	O
on	O
then	O
the	O
same	O
is	O
true	O
for	O
,	O
and	O
Moreover	O
,	O
if	O
and	O
are	O
both	O
Riemann-integrable	O
then	O
is	O
also	O
Riemann-integrable	O
,	O
and	O
This	O
inequality	O
,	O
known	O
as	O
the	O
Cauchy	O
–	O
Schwarz	O
inequality	O
,	O
plays	O
a	O
prominent	O
role	O
in	O
Hilbert	O
space	O
theory	O
,	O
where	O
the	O
left	O
hand	O
side	O
is	O
interpreted	O
as	O
the	O
inner	O
product	O
of	O
two	O
square-integrable	B-Algorithm
functions	I-Algorithm
and	O
on	O
the	O
interval	O
.	O
</s>
<s>
The	O
values	O
and	O
,	O
the	O
end-points	O
of	O
the	O
interval	O
,	O
are	O
called	O
the	O
limits	B-Algorithm
of	O
integration	O
of	O
.	O
</s>
<s>
Integrals	B-Algorithm
can	O
also	O
be	O
defined	O
if	O
:	O
</s>
<s>
The	O
first	O
convention	O
is	O
necessary	O
in	O
consideration	O
of	O
taking	O
integrals	B-Algorithm
over	O
subintervals	O
of	O
;	O
the	O
second	O
says	O
that	O
an	O
integral	O
taken	O
over	O
a	O
degenerate	O
interval	O
,	O
or	O
a	O
point	O
,	O
should	O
be	O
zero	O
.	O
</s>
<s>
One	O
reason	O
for	O
the	O
first	O
convention	O
is	O
that	O
the	O
integrability	O
of	O
on	O
an	O
interval	O
implies	O
that	O
is	O
integrable	O
on	O
any	O
subinterval	O
,	O
but	O
in	O
particular	O
integrals	B-Algorithm
have	O
the	O
property	O
that	O
if	O
is	O
any	O
element	O
of	O
,	O
then	O
:	O
</s>
<s>
The	O
fundamental	O
theorem	O
of	O
calculus	O
is	O
the	O
statement	O
that	O
differentiation	B-Algorithm
and	O
integration	O
are	O
inverse	O
operations	O
:	O
if	O
a	O
continuous	O
function	O
is	O
first	O
integrated	O
and	O
then	O
differentiated	O
,	O
the	O
original	O
function	O
is	O
retrieved	O
.	O
</s>
<s>
An	O
important	O
consequence	O
,	O
sometimes	O
called	O
the	O
second	O
fundamental	O
theorem	O
of	O
calculus	O
,	O
allows	O
one	O
to	O
compute	O
integrals	B-Algorithm
by	O
using	O
an	O
antiderivative	B-Algorithm
of	O
the	O
function	O
to	O
be	O
integrated	O
.	O
</s>
<s>
Let	O
be	O
a	O
real-valued	O
function	O
defined	O
on	O
a	O
closed	O
interval	O
 [  ] 	O
that	O
admits	O
an	O
antiderivative	B-Algorithm
on	O
.	O
</s>
<s>
A	O
"	O
proper	O
"	O
Riemann	O
integral	O
assumes	O
the	O
integrand	O
is	O
defined	O
and	O
finite	O
on	O
a	O
closed	O
and	O
bounded	B-Algorithm
interval	O
,	O
bracketed	O
by	O
the	O
limits	B-Algorithm
of	O
integration	O
.	O
</s>
<s>
In	O
some	O
cases	O
such	O
integrals	B-Algorithm
may	O
be	O
defined	O
by	O
considering	O
the	O
limit	B-Algorithm
of	O
a	O
sequence	O
of	O
proper	O
Riemann	O
integrals	B-Algorithm
on	O
progressively	O
larger	O
intervals	O
.	O
</s>
<s>
If	O
the	O
interval	O
is	O
unbounded	O
,	O
for	O
instance	O
at	O
its	O
upper	O
end	O
,	O
then	O
the	O
improper	O
integral	O
is	O
the	O
limit	B-Algorithm
as	O
that	O
endpoint	O
goes	O
to	O
infinity	O
:	O
</s>
<s>
If	O
the	O
integrand	O
is	O
only	O
defined	O
or	O
finite	O
on	O
a	O
half-open	O
interval	O
,	O
for	O
instance	O
,	O
then	O
again	O
a	O
limit	B-Algorithm
may	O
provide	O
a	O
finite	O
result	O
:	O
</s>
<s>
That	O
is	O
,	O
the	O
improper	O
integral	O
is	O
the	O
limit	B-Algorithm
of	O
proper	O
integrals	B-Algorithm
as	O
one	O
endpoint	O
of	O
the	O
interval	O
of	O
integration	O
approaches	O
either	O
a	O
specified	O
real	O
number	O
,	O
or	O
,	O
or	O
.	O
</s>
<s>
In	O
more	O
complicated	O
cases	O
,	O
limits	B-Algorithm
are	O
required	O
at	O
both	O
endpoints	O
,	O
or	O
at	O
interior	O
points	O
.	O
</s>
<s>
Just	O
as	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
a	O
positive	O
function	O
of	O
one	O
variable	O
represents	O
the	O
area	O
of	O
the	O
region	O
between	O
the	O
graph	B-Application
of	O
the	O
function	O
and	O
the	O
x-axis	O
,	O
the	O
double	O
integral	O
of	O
a	O
positive	O
function	O
of	O
two	O
variables	O
represents	O
the	O
volume	O
of	O
the	O
region	O
between	O
the	O
surface	O
defined	O
by	O
the	O
function	O
and	O
the	O
plane	O
that	O
contains	O
its	O
domain	O
.	O
</s>
<s>
This	O
reduces	O
the	O
problem	O
of	O
computing	O
a	O
double	O
integral	O
to	O
computing	O
one-dimensional	O
integrals	B-Algorithm
.	O
</s>
<s>
Such	O
integrals	B-Algorithm
are	O
known	O
as	O
line	O
integrals	B-Algorithm
and	O
surface	O
integrals	B-Algorithm
respectively	O
.	O
</s>
<s>
Various	O
different	O
line	O
integrals	B-Algorithm
are	O
in	O
use	O
.	O
</s>
<s>
This	O
weighting	O
distinguishes	O
the	O
line	O
integral	O
from	O
simpler	O
integrals	B-Algorithm
defined	O
on	O
intervals	O
.	O
</s>
<s>
Many	O
simple	O
formulas	O
in	O
physics	O
have	O
natural	O
continuous	O
analogs	O
in	O
terms	O
of	O
line	O
integrals	B-Algorithm
;	O
for	O
example	O
,	O
the	O
fact	O
that	O
work	O
is	O
equal	O
to	O
force	O
,	O
,	O
multiplied	O
by	O
displacement	O
,	O
,	O
may	O
be	O
expressed	O
(	O
in	O
terms	O
of	O
vector	O
quantities	O
)	O
as	O
:	O
</s>
<s>
A	O
surface	O
integral	O
generalizes	O
double	O
integrals	B-Algorithm
to	O
integration	O
over	O
a	O
surface	O
(	O
which	O
may	O
be	O
a	O
curved	O
set	O
in	O
space	O
)	O
;	O
it	O
can	O
be	O
thought	O
of	O
as	O
the	O
double	O
integral	O
analog	O
of	O
the	O
line	O
integral	O
.	O
</s>
<s>
For	O
an	O
example	O
of	O
applications	O
of	O
surface	O
integrals	B-Algorithm
,	O
consider	O
a	O
vector	O
field	O
on	O
a	O
surface	O
;	O
that	O
is	O
,	O
for	O
each	O
point	O
in	O
,	O
is	O
a	O
vector	O
.	O
</s>
<s>
The	O
flux	B-General_Concept
is	O
defined	O
as	O
the	O
quantity	O
of	O
fluid	O
flowing	O
through	O
in	O
unit	O
amount	O
of	O
time	O
.	O
</s>
<s>
To	O
find	O
the	O
flux	B-General_Concept
,	O
one	O
need	O
to	O
take	O
the	O
dot	O
product	O
of	O
with	O
the	O
unit	O
surface	O
normal	O
to	O
at	O
each	O
point	O
,	O
which	O
will	O
give	O
a	O
scalar	O
field	O
,	O
which	O
is	O
integrated	O
over	O
the	O
surface	O
:	O
</s>
<s>
The	O
fluid	O
flux	B-General_Concept
in	O
this	O
example	O
may	O
be	O
from	O
a	O
physical	O
fluid	O
such	O
as	O
water	O
or	O
air	O
,	O
or	O
from	O
electrical	O
or	O
magnetic	O
flux	B-General_Concept
.	O
</s>
<s>
Thus	O
surface	O
integrals	B-Algorithm
have	O
applications	O
in	O
physics	O
,	O
particularly	O
with	O
the	O
classical	O
theory	O
of	O
electromagnetism	O
.	O
</s>
<s>
A	O
differential	O
form	O
is	O
a	O
mathematical	O
concept	O
in	O
the	O
fields	O
of	O
multivariable	O
calculus	O
,	O
differential	O
topology	O
,	O
and	O
tensors	B-Device
.	O
</s>
<s>
For	O
example	O
,	O
a	O
one-form	B-Algorithm
is	O
a	O
weighted	O
sum	O
of	O
the	O
differentials	O
of	O
the	O
coordinates	O
,	O
such	O
as	O
:	O
</s>
<s>
A	O
differential	O
one-form	B-Algorithm
can	O
be	O
integrated	O
over	O
an	O
oriented	O
path	O
,	O
and	O
the	O
resulting	O
integral	O
is	O
just	O
another	O
way	O
of	O
writing	O
a	O
line	O
integral	O
.	O
</s>
<s>
A	O
two-form	O
can	O
be	O
integrated	O
over	O
an	O
oriented	O
surface	O
,	O
and	O
the	O
resulting	O
integral	O
is	O
equivalent	O
to	O
the	O
surface	O
integral	O
giving	O
the	O
flux	B-General_Concept
of	O
.	O
</s>
<s>
The	O
exterior	O
derivative	B-Algorithm
plays	O
the	O
role	O
of	O
the	O
gradient	O
and	O
curl	O
of	O
vector	O
calculus	O
,	O
and	O
Stokes	O
 '	O
theorem	O
simultaneously	O
generalizes	O
the	O
three	O
theorems	O
of	O
vector	O
calculus	O
:	O
the	O
divergence	O
theorem	O
,	O
Green	O
's	O
theorem	O
,	O
and	O
the	O
Kelvin-Stokes	O
theorem	O
.	O
</s>
<s>
Summations	O
and	O
integrals	B-Algorithm
can	O
be	O
put	O
on	O
the	O
same	O
foundations	O
using	O
the	O
theory	O
of	O
Lebesgue	O
integrals	B-Algorithm
or	O
time-scale	O
calculus	O
.	O
</s>
<s>
An	O
integration	O
that	O
is	O
performed	O
not	O
over	O
a	O
variable	O
(	O
or	O
,	O
in	O
physics	O
,	O
over	O
a	O
space	O
or	O
time	O
dimension	O
)	O
,	O
but	O
over	O
a	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
,	O
is	O
referred	O
to	O
as	O
a	O
functional	O
integral	O
.	O
</s>
<s>
Integrals	B-Algorithm
are	O
used	O
extensively	O
in	O
many	O
areas	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
probability	O
theory	O
,	O
integrals	B-Algorithm
are	O
used	O
to	O
determine	O
the	O
probability	O
of	O
some	O
random	O
variable	O
falling	O
within	O
a	O
certain	O
range	O
.	O
</s>
<s>
Integrals	B-Algorithm
can	O
be	O
used	O
for	O
computing	O
the	O
area	O
of	O
a	O
two-dimensional	O
region	O
that	O
has	O
a	O
curved	O
boundary	O
,	O
as	O
well	O
as	O
computing	O
the	O
volume	O
of	O
a	O
three-dimensional	O
object	O
that	O
has	O
a	O
curved	O
boundary	O
.	O
</s>
<s>
The	O
area	O
of	O
a	O
two-dimensional	O
region	O
can	O
be	O
calculated	O
using	O
the	O
aforementioned	O
definite	B-Algorithm
integral	I-Algorithm
.	O
</s>
<s>
Using	O
an	O
integral	O
with	O
bounds	O
and	O
,	O
the	O
volume	O
of	O
the	O
disc	O
is	O
equal	O
to:Integrals	O
are	O
also	O
used	O
in	O
physics	O
,	O
in	O
areas	O
like	O
kinematics	O
to	O
find	O
quantities	O
like	O
displacement	O
,	O
time	O
,	O
and	O
velocity	O
.	O
</s>
<s>
Integrals	B-Algorithm
are	O
also	O
used	O
in	O
thermodynamics	O
,	O
where	O
thermodynamic	O
integration	O
is	O
used	O
to	O
calculate	O
the	O
difference	O
in	O
free	O
energy	O
between	O
two	O
given	O
states	O
.	O
</s>
<s>
The	O
most	O
basic	O
technique	O
for	O
computing	O
definite	B-Algorithm
integrals	I-Algorithm
of	O
one	O
real	O
variable	O
is	O
based	O
on	O
the	O
fundamental	O
theorem	O
of	O
calculus	O
.	O
</s>
<s>
Then	O
,	O
find	O
an	O
antiderivative	B-Algorithm
of	O
;	O
that	O
is	O
,	O
a	O
function	O
such	O
that	O
on	O
the	O
interval	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
necessary	O
to	O
use	O
one	O
of	O
the	O
many	O
techniques	O
that	O
have	O
been	O
developed	O
to	O
evaluate	O
integrals	B-Algorithm
.	O
</s>
<s>
Alternative	O
methods	O
exist	O
to	O
compute	O
more	O
complex	O
integrals	B-Algorithm
.	O
</s>
<s>
Many	O
nonelementary	O
integrals	B-Algorithm
can	O
be	O
expanded	O
in	O
a	O
Taylor	O
series	O
and	O
integrated	O
term	O
by	O
term	O
.	O
</s>
<s>
There	O
are	O
also	O
many	O
less	O
common	O
ways	O
of	O
calculating	O
definite	B-Algorithm
integrals	I-Algorithm
;	O
for	O
instance	O
,	O
Parseval	O
's	O
identity	O
can	O
be	O
used	O
to	O
transform	O
an	O
integral	O
over	O
a	O
rectangular	O
region	O
into	O
an	O
infinite	O
sum	O
.	O
</s>
<s>
Specific	O
results	O
which	O
have	O
been	O
worked	O
out	O
by	O
various	O
techniques	O
are	O
collected	O
in	O
the	O
list	O
of	O
integrals	B-Algorithm
.	O
</s>
<s>
Extensive	O
tables	O
of	O
integrals	B-Algorithm
have	O
been	O
compiled	O
and	O
published	O
over	O
the	O
years	O
for	O
this	O
purpose	O
.	O
</s>
<s>
With	O
the	O
spread	O
of	O
computers	O
,	O
many	O
professionals	O
,	O
educators	O
,	O
and	O
students	O
have	O
turned	O
to	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
that	O
are	O
specifically	O
designed	O
to	O
perform	O
difficult	O
or	O
tedious	O
tasks	O
,	O
including	O
integration	O
.	O
</s>
<s>
Symbolic	O
integration	O
has	O
been	O
one	O
of	O
the	O
motivations	O
for	O
the	O
development	O
of	O
the	O
first	O
such	O
systems	O
,	O
like	O
Macsyma	B-Language
and	O
Maple	B-Language
.	O
</s>
<s>
A	O
major	O
mathematical	O
difficulty	O
in	O
symbolic	O
integration	O
is	O
that	O
in	O
many	O
cases	O
,	O
a	O
relatively	O
simple	O
function	O
does	O
not	O
have	O
integrals	B-Algorithm
that	O
can	O
be	O
expressed	O
in	O
closed	O
form	O
involving	O
only	O
elementary	O
functions	O
,	O
include	O
rational	O
and	O
exponential	O
functions	O
,	O
logarithm	O
,	O
trigonometric	O
functions	O
and	O
inverse	O
trigonometric	O
functions	O
,	O
and	O
the	O
operations	O
of	O
multiplication	O
and	O
composition	O
.	O
</s>
<s>
The	O
Risch	O
algorithm	O
provides	O
a	O
general	O
criterion	O
to	O
determine	O
whether	O
the	O
antiderivative	B-Algorithm
of	O
an	O
elementary	O
function	O
is	O
elementary	O
and	O
to	O
compute	O
the	O
integral	O
if	O
is	O
elementary	O
.	O
</s>
<s>
However	O
,	O
functions	O
with	O
closed	O
expressions	O
of	O
antiderivatives	B-Algorithm
are	O
the	O
exception	O
,	O
and	O
consequently	O
,	O
computerized	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
have	O
no	O
hope	O
of	O
being	O
able	O
to	O
find	O
an	O
antiderivative	B-Algorithm
for	O
a	O
randomly	O
constructed	O
elementary	O
function	O
.	O
</s>
<s>
On	O
the	O
positive	O
side	O
,	O
if	O
the	O
'	O
building	O
blocks	O
 '	O
for	O
antiderivatives	B-Algorithm
are	O
fixed	O
in	O
advance	O
,	O
it	O
may	O
still	O
be	O
possible	O
to	O
decide	O
whether	O
the	O
antiderivative	B-Algorithm
of	O
a	O
given	O
function	O
can	O
be	O
expressed	O
using	O
these	O
blocks	O
and	O
operations	O
of	O
multiplication	O
and	O
composition	O
and	O
to	O
find	O
the	O
symbolic	O
answer	O
whenever	O
it	O
exists	O
.	O
</s>
<s>
The	O
Risch	O
algorithm	O
,	O
implemented	O
in	O
Mathematica	B-Language
,	O
Maple	B-Language
and	O
other	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
,	O
does	O
just	O
that	O
for	O
functions	O
and	O
antiderivatives	B-Algorithm
built	O
from	O
rational	O
functions	O
,	O
radicals	O
,	O
logarithm	O
,	O
and	O
exponential	O
functions	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
may	O
be	O
useful	O
to	O
have	O
,	O
in	O
the	O
set	O
of	O
antiderivatives	B-Algorithm
,	O
the	O
special	O
functions	O
(	O
like	O
the	O
Legendre	O
functions	O
,	O
the	O
hypergeometric	O
function	O
,	O
the	O
gamma	O
function	O
,	O
the	O
incomplete	O
gamma	O
function	O
and	O
so	O
on	O
)	O
.	O
</s>
<s>
This	O
provides	O
an	O
algorithm	O
to	O
express	O
the	O
antiderivative	B-Algorithm
of	O
a	O
D-finite	O
function	O
as	O
the	O
solution	O
of	O
a	O
differential	O
equation	O
.	O
</s>
<s>
This	O
theory	O
also	O
allows	O
one	O
to	O
compute	O
the	O
definite	B-Algorithm
integral	I-Algorithm
of	O
a	O
D-function	O
as	O
the	O
sum	O
of	O
a	O
series	O
given	O
by	O
the	O
first	O
coefficients	O
and	O
provides	O
an	O
algorithm	O
to	O
compute	O
any	O
coefficient	O
.	O
</s>
<s>
Rubi	O
,	O
a	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
rule-based	O
integrator	O
,	O
pattern	O
matches	O
an	O
extensive	O
system	O
of	O
symbolic	O
integration	O
rules	O
to	O
integrate	O
a	O
wide	O
variety	O
of	O
integrands	O
.	O
</s>
<s>
This	O
system	O
uses	O
over	O
6600	O
integration	O
rules	O
to	O
compute	O
integrals	B-Algorithm
.	O
</s>
<s>
The	O
method	O
of	O
brackets	O
is	O
a	O
generalization	O
of	O
Ramanujan	O
's	O
master	O
theorem	O
that	O
can	O
be	O
applied	O
to	O
a	O
wide	O
range	O
of	O
univariate	O
and	O
multivariate	O
integrals	B-Algorithm
.	O
</s>
<s>
Definite	B-Algorithm
integrals	I-Algorithm
may	O
be	O
approximated	O
using	O
several	O
methods	O
of	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
A	O
better	O
approach	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
replaces	O
the	O
rectangles	O
used	O
in	O
a	O
Riemann	O
sum	O
with	O
trapezoids	O
.	O
</s>
<s>
The	O
trapezoidal	B-Algorithm
rule	I-Algorithm
weights	O
the	O
first	O
and	O
last	O
values	O
by	O
one	O
half	O
,	O
then	O
multiplies	O
by	O
the	O
step	O
width	O
to	O
obtain	O
a	O
better	O
approximation	O
.	O
</s>
<s>
The	O
idea	O
behind	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
that	O
more	O
accurate	O
approximations	O
to	O
the	O
function	O
yield	O
better	O
approximations	O
to	O
the	O
integral	O
,	O
can	O
be	O
carried	O
further	O
:	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
approximates	O
the	O
integrand	O
by	O
a	O
piecewise	O
quadratic	O
function	O
.	O
</s>
<s>
Riemann	O
sums	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
and	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
are	O
examples	O
of	O
a	O
family	O
of	O
quadrature	B-Algorithm
rules	I-Algorithm
called	O
the	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
.	O
</s>
<s>
The	O
degree	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
quadrature	I-Algorithm
rule	I-Algorithm
approximates	O
the	O
polynomial	O
on	O
each	O
subinterval	O
by	O
a	O
degree	O
polynomial	O
.	O
</s>
<s>
This	O
polynomial	O
is	O
chosen	O
to	O
interpolate	B-Algorithm
the	O
values	O
of	O
the	O
function	O
on	O
the	O
interval	O
.	O
</s>
<s>
One	O
solution	O
to	O
this	O
problem	O
is	O
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
,	O
in	O
which	O
the	O
integrand	O
is	O
approximated	O
by	O
expanding	O
it	O
in	O
terms	O
of	O
Chebyshev	O
polynomials	O
.	O
</s>
<s>
Romberg	B-Algorithm
's	I-Algorithm
method	I-Algorithm
halves	O
the	O
step	O
widths	O
incrementally	O
,	O
giving	O
trapezoid	O
approximations	O
denoted	O
by	O
,	O
,	O
and	O
so	O
on	O
,	O
where	O
is	O
half	O
of	O
.	O
</s>
<s>
It	O
then	O
interpolate	B-Algorithm
a	O
polynomial	O
through	O
the	O
approximations	O
,	O
and	O
extrapolate	O
to	O
.	O
</s>
<s>
Gaussian	B-Algorithm
quadrature	I-Algorithm
evaluates	O
the	O
function	O
at	O
the	O
roots	O
of	O
a	O
set	O
of	O
orthogonal	O
polynomials	O
.	O
</s>
<s>
The	O
computation	O
of	O
higher-dimensional	O
integrals	B-Algorithm
(	O
for	O
example	O
,	O
volume	O
calculations	O
)	O
makes	O
important	O
use	O
of	O
such	O
alternatives	O
as	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
Kempf	O
,	O
Jackson	O
and	O
Morales	O
demonstrated	O
mathematical	O
relations	O
that	O
allow	O
an	O
integral	O
to	O
be	O
calculated	O
by	O
means	O
of	O
differentiation	B-Algorithm
.	O
</s>
<s>
Their	O
calculus	O
involves	O
the	O
Dirac	O
delta	O
function	O
and	O
the	O
partial	O
derivative	B-Algorithm
operator	O
.	O
</s>
<s>
This	O
can	O
also	O
be	O
applied	O
to	O
functional	O
integrals	B-Algorithm
,	O
allowing	O
them	O
to	O
be	O
computed	O
by	O
functional	O
differentiation	B-Algorithm
.	O
</s>
