View source for PlanetPhysics/Fundamental Theorem of Integral Calculus

Consider the sequence of numbers <math>\{f_0,f_1,...f_N\}</math> and define the difference <math>\Delta f_j = f_j-f_{j-1}</math>.  Now sum the differences and not that all but the first and last terms cancel:

<math>\sum \Delta f_j = (f_1-f_0) + (f_2-f_1) + (f_3-f_2) + ... + (f_{N-2} - f_{N-1}) + (f_N - f_{N-1}) = f_N-f_0</math>

In other words <math>\int_a^bdf = f(b)-f(a)</math>.  It seems obvious that,

<math>\int_a^b\frac{df}{dx}dx= \int df = f(b)-f(a)</math>

Changing variables:

<math>\int_a^x\frac{df}{ds}= \int df = f(x)-f(a)</math>

or as an indefinite integral:

<math>\int f'(x)dx = f(x) + C</math>

====Converse====
In other words, the integral of the derivative of a function is the original function.  But what of the derivative of the integral?  Let,

<math>g(x) = \int_a^x f(s)ds = \sum_0^N f_j \Delta x = \left(f_0+f_1+...+f_N \right)\Delta x</math> where <math>f_N = f(x)</math>.

Here, we assume that all the intervals <math>\Delta x</math> in the [[w:Riemann sum|Riemann sum]] are equal. To find <math>g(x+\Delta x)</math> we need to add one extra term to the Riemann sum:

<math>g(x+\Delta x) = \int_a^{x+\Delta x} f(s)ds </math>
::<math>= \sum_0^{N+1} f_j \Delta x = \underbrace{\sum_0^{N} f_j \Delta x}_{g(x)} + f_{N+1}\Delta x</math>.

[[File:FTC_geometric.svg|240px|thumb|As shown in red, the change in area (&int;fdx) of a function is closely related to the value of the function, ''f(x)'' at the point where ''x'' changes to {{nowrap|''x+&delta;x''}}]]
<math>g(x+\Delta x) = g(x) + f(x+\Delta x)\Delta x</math>.

Rearrange this to obtain:

<math>f(x+\Delta x)\Delta x = g(x+\Delta x) - g(x) </math>

<small>The original Planet Physics version still remains on the page as a hidden comment (visible in "edit" mode)</small>
<!--The derivative of a real [[../Bijective/|function]], which has on a whole interval a constant value <math>c</math>, vanishes in every point of this interval:
<math>\frac{d}{dx}c \;=\; 0</math>\\

The converse [[../Formula/|theorem]] of this is also true.\, Ernst Lindel\"of calls it the ''fundamental theorem of integral calculus''  (in Finnish ''integraalilaskun peruslause'' ).\, It can be formulated as

'''Theorem.''' \, If a real function in continuous and its derivative vanishes in all points of an interval, the value of this function does not change on this interval.

''Proof.'' \, We make the antithesis that there were on the interval two distinct points <math>x_1</math> and <math>x_2</math> with\, <math>f(x_1) \neq f(x_2)</math>.\, Then the mean-value theorem guarantees a point <math>\xi</math> between <math>x_1</math> and <math>x_2</math> such that
<math>f'(\xi) \;=\; \frac{f(x_1)\!-\!f(x_2)}{x_1\!-\!x_2},</math>
which value is distinct from zero.\, This is, however, impossible by the assumption of the theorem.\, So the antithesis is wrong and the theorem right.\\

The contents of the theorem may be expressed also such that if two functions have the same derivative on a whole interval, then the difference of the functions is constant on this interval.\, Accordingly, if <math>F</math> is an antiderivative of a function <math>f</math>, then any other antiderivative of <math>f</math> has the form <math>x \mapsto F(x)\!+\!C</math>, where <math>C</math> is a constant.-->

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