Editing Calculus (section)

=== Integral calculus ===
{{Main|Integral}}
{{multiple image| total_width = 300px | direction = vertical
| image1 = Integral as region under curve.svg
| caption1 = Integration can be thought of as measuring the area under a curve, defined by {{math|''f''(''x'')}}, between two points (here {{math|'' a''}} and {{math|''b''}}).
| image2 = Riemann integral regular.gif
| caption2 = A sequence of midpoint Riemann sums over a regular partition of an interval: the total area of the rectangles converges to the integral of the function.
}}
''Integral calculus'' is the study of the definitions, properties, and applications of two related concepts, the ''indefinite integral'' and the ''definite integral''. The process of finding the value of an integral is called ''integration''.<ref name=":5">{{Cite book |last1=Herman |first1=Edwin |url=https://openstax.org/details/books/calculus-volume-1 |title=Calculus |volume=1 |last2=Strang |first2=Gilbert |date=2017 |publisher=OpenStax |isbn=978-1-938168-02-4 |location=Houston, Texas |oclc=1022848630 |display-authors=etal |author-link2=Gilbert Strang |access-date=26 July 2022 |archive-date=23 September 2022 |archive-url=https://web.archive.org/web/20220923230919/https://openstax.org/details/books/calculus-volume-1 |url-status=live }}</ref>{{Rp|page=508}} The indefinite integral, also known as the ''[[antiderivative]]'', is the inverse operation to the derivative.<ref name=":4" />{{Rp|pages=163–165}} {{math|''F''}} is an indefinite integral of {{math|''f''}} when {{math|''f''}} is a derivative of {{math|''F''}}.  (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the [[x-axis]]. The technical definition of the definite integral involves the [[limit (mathematics)|limit]] of a sum of areas of rectangles, called a [[Riemann sum]].<ref name=":2">{{Cite book |last1=Hughes-Hallett |first1=Deborah |title=Calculus: Single and Multivariable |last2=McCallum |first2=William G. |last3=Gleason |first3=Andrew M. |last4=Connally |first4=Eric |date=2013 |publisher=Wiley |isbn=978-0-470-88861-2 |edition=6th |location=Hoboken, NJ |oclc=794034942 |display-authors=3 |author-link=Deborah Hughes Hallett |author-link2=William G. McCallum|author-link3=Andrew M. Gleason}}</ref>{{Rp|page=282}}

A motivating example is the distance traveled in a given time.<ref name=":4" />{{Rp|pages=153}} If the speed is constant, only multiplication is needed:
:<math>\mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time}</math>
But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a [[Riemann sum]]) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.

When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time.  For example, traveling a steady 50&nbsp;mph for 3 hours results in a total distance of 150 miles.  Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed.  Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.<ref name=":5"/>{{rp|535}}  This connection between the area under a curve and the distance traveled can be extended to ''any'' irregularly shaped region exhibiting a fluctuating velocity over a given period. If {{math|''f''(''x'')}} represents speed as it varies over time, the distance traveled between the times represented by {{math|'' a''}} and {{math|''b''}} is the area of the region between {{math|''f''(''x'')}} and the {{math|''x''}}-axis, between {{math|''x'' {{=}} ''a''}} and {{math|''x'' {{=}} ''b''}}.

To approximate that area, an intuitive method would be to divide up the distance between {{math|'' a''}} and {{math|''b''}} into several equal segments, the length of each segment represented by the symbol {{math|Δ''x''}}. For each small segment, we can choose one value of the function {{math|''f''(''x'')}}. Call that value {{math|''h''}}. Then the area of the rectangle with base {{math|Δ''x''}} and height {{math|''h''}} gives the distance (time {{math|Δ''x''}} multiplied by speed {{math|''h''}}) traveled in that segment.   Associated with each segment is the average value of the function above it, {{math|''f''(''x'') {{=}} ''h''}}. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for {{math|Δ''x''}} will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit as {{math|Δ''x''}} approaches zero.<ref name=":5"/>{{rp|512–522}}

The symbol of integration is <math>\int </math>, an [[long s|elongated ''S'']] chosen to suggest summation.<ref name=":5" />{{Rp|pages=529}} The definite integral is written as:

:<math>\int_a^b f(x)\, dx</math>

and is read "the integral from ''a'' to ''b'' of ''f''-of-''x'' with respect to ''x''." The Leibniz notation {{math|''dx''}} is intended to suggest dividing the area under the curve into an infinite number of rectangles so that their width {{math|Δ''x''}} becomes the infinitesimally small {{math|''dx''}}.<ref name="TMU"/>{{Rp|44}}

The indefinite integral, or antiderivative, is written:

:<math>\int f(x)\, dx.</math>

Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is a family of functions differing only by a constant.<ref name=":2" />{{Rp|page=326}} Since the derivative of the function {{math|''y'' {{=}} ''x''<sup>2</sup> + ''C''}}, where {{math|''C''}} is any constant, is {{math|''y′'' {{=}} 2''x''}}, the antiderivative of the latter is given by:
:<math>\int 2x\, dx = x^2 + C.</math>
The unspecified constant {{math|''C''}} present in the indefinite integral or antiderivative is known as the [[constant of integration]].<ref>{{cite book|first1=William |last1=Moebs |first2=Samuel J. |last2=Ling |first3=Jeff |last3=Sanny |display-authors=etal |title=University Physics, Volume 1 |publisher=OpenStax |year=2022 |isbn=978-1-947172-20-3 |oclc=961352944}}</ref>{{rp|135}}