This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Integral


00:00:46 1 History
00:03:20 1.1 Pre-calculus integration
00:03:28 1.2 Newton and Leibniz
00:04:47 1.3 Formalization
00:05:13 1.4 Historical notation
00:05:22 2 Applications
00:06:11 3 Terminology and notation
00:06:20 3.1 Standard
00:07:23 3.2 Meaning of the symbol idx/i
00:08:37 3.3 Variants
00:14:36 4 Interpretations of the integral
00:14:46 5 Formal definitions
00:16:56 5.1 Riemann integral
00:19:43 5.2 Lebesgue integral
00:21:36 5.3 Other integrals
00:22:19 6 Properties
00:22:31 6.1 Linearity
00:23:13 6.2 Inequalities
00:23:24 6.3 Conventions
00:23:34 7 Fundamental theorem of calculus
00:27:26 7.1 Statements of theorems
00:27:56 7.1.1 Fundamental theorem of calculus
00:30:46 7.1.2 Second fundamental theorem of calculus
00:33:06 7.2 Calculating integrals
00:35:44 8 Extensions
00:37:23 8.1 Improper integrals
00:37:32 8.2 Multiple integration
00:41:43 8.3 Line integrals
00:47:38 8.4 Surface integrals
00:49:08 8.5 Contour integrals
00:49:28 8.6 Integrals of differential forms
00:52:48 8.7 Summations
00:53:21 9 Computation
00:53:30 9.1 Analytical
00:54:28 9.2 Symbolic
00:55:27 9.3 Numerical
00:57:27 9.4 Mechanical
00:57:56 9.5 Geometrical
00:58:05 10 See also
01:00:17 11 Notes
01:03:12 12 References
01:05:12 13 External links
01:06:44 13.1 Online books
01:07:28 Integrals of differential forms
01:11:17 Summations
01:11:37 Computation
01:11:46 Analytical
01:14:41 Symbolic
01:17:56 Numerical
01:22:46 0...2
01:25:26 0. However, the substitution u
01:26:03 Mechanical
01:26:26 Geometrical
01:26:42 See also



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SUMMARY
=======
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral





∫

a


b


f
(
x
)

d
x


{\displaystyle \int _{a}^{b}f(x)\,dx}
is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.
The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:




F
(
x
)
=
∫
f
(
x
)

d
x
.


{\displaystyle F(x)=\int f(x)\,dx.}
The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by





∫

a


b



f
(
x
)
d
x
=


[

F
(
x
)

]


a


b


=
F
(
b
)
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