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This is [[April Fool's Day]], [[2002]]

April fools hoaxes for this year:
* [[The Register]]: [http://www.theregister.co.uk/content/28/24652.html reported] [[AOL]] buying up [[weblog]]s
* The [[Open Directory]] [http://[email protected]/April-01-2002_GOD_PR.html re-branded] itself as the 'Microsoft Directory Project'
* [[Kuro5hin]] [http://www.kuro5hin.org/story/2002/4/1/05035/09001 acquired] met4filter.org
* [[Slashdot]] [http://slashdot.org/articles/02/04/01/1452256.shtml?tid=124 announced] that is will start posting advertiser-sponsored news stories, and disable anonymous posting.
* [[Google]] [http://www.google.com/technology/pigeonrank.html described] its PigeonRank system.
* [[The Guardian]] [http://www.guardian.co.uk/Archive/Article/0,4273,4385266,00.html profiled] Harmony Cousins
* [[CPAN]] [http://www.cpan.org/ renamed] itself the 'Comprehensive Java Archive Network'
* The annual spoof [[Linus Torvalds]] post on the [[Linux kernel mailing list]] [http://www.cs.helsinki.fi/linux/linux-kernel/2002-13/0063.html announced] his resignation from the Linux effort
* [[IETF]] published [[April 1st RFC]]s RFC3251 (Distribution of electricity over IP) and RFC3252 (Encapsulating IP in XML).
* [[MIT]] [http://www.mit.edu changed] its home page to a spoof of the [[Google]] home page.
* a spoof Yahoo News story was [http://news.yahoo.com:PK%20Interactive%20Receives%20Funding%20from%[email protected]/Yahoo!%20News%20-%20PK%20Interactive%20receives%20funding%20from%20idealab.htm circulated] stating that PK Interactive had received funding from [[idealab]] (note: check the URL below closely)
* The TidBITS newsletter [http://www.tidbits.com/tb-issues/TidBITS-623.html offered] a spoof issue


''Please add more [[2002]] April Fool hoaxes here, including any relevant external links''{{Calculus}}
The '''fundamental theorem of calculus''' is the statement that the two central operations of [[calculus]], [[derivative|differentiation]] and [[integral|integration]], are inverses of each other. This means that if a [[continuous]] [[function (mathematics)|function]] is first integrated and then differentiated, the original function is retrieved. This theorem is of such central importance in calculus that it deserves to be called the [[fundamental theorem]] for the entire field of study. An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an [[antiderivative]] of the function to be integrated. James Stewart (2003, 394) of [[McMaster University]] credits the English mathematician [[Isaac Barrow]] for originating the idea that led to the fundamental theorem.

== Intuition ==
Intuitively, the theorem simply says that the sum of [[infinitesimal]] changes in a quantity over time (or some other quantity) add up to the net change in quantity.

To get a feeling for the statement, we will start with an example. Suppose a particle travels in a straight line with its position given by ''x''(''t'') where ''t'' is time.  The derivative of this function is equal to the infinitesimal change in ''x'' per infinitesimal change in time (of course, the derivative itself is dependent on time).  Let us define this change in distance per time as the speed ''v'' of the particle.  In [[Leibniz notation|Leibniz's notation]]:

:<math>\frac{dx}{dt} = v(t) </math>

Rearranging that equation, it is clear that:

:<math>dx = v(t)\,dt </math>

By the logic above, a change in ''x'', call it <math>\Delta x</math>, is the sum of the infinitesimal changes ''dx''.  It is also equal to the sum of the infinitesimal products of the derivative and time.  This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative.  Clearly, this operation works in reverse as we can differentiate the result of our integral to recover the speed function.

== Proof ==
This is a limit proof by [[Riemann integral|Riemann Sums]].

<math>\int_{b}^{a} f(x)\,dx = F(b) - F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>.

Have <math>a = x_0 < x_1 < x_2 < \ldots < x_{n-1} < x_n = b</math>, so that:

<math>F(b) - F(a) = F(x_n) - F(x_0) \,</math>

Now, we add and subtract the same quantity so that:
<math>F(x_n) - F(x_0) = F(x_n) - F(x_{n-1}) + F(x_{n-1}) - \ldots - F(x_1) + F(x_1) - F(x_0)</math>

<math>= \sum_{i=1}^n [F(x_i) - F(x_{i-1})]</math>

<blockquote style="padding: 1em; border: medium dotted">
<p>Here we employ the [[Mean Value Theorem]].  In brief,</p>

<math>f'(c) = \frac{f(b) - f(a)}{b - a}  </math>

<math>  f'(c)(b - a) = f(b) - f(a) \,</math>

</blockquote>

[[Image:Riemann.gif|300px|right|frame|A converging sequence of Riemann sums.  The numbers in the upper right are the areas of the grey rectangles.  They converge to the integral of the function.]]

<math>= \sum_{i=1}^n [F'(c_i)(x_i - x_{i-1})]</math>

As the derivative of the antiderivative is the original function, <math>F'(c_i) = f(c_i)</math>.  Also, <math>x_i - x_{i-1}</math> can be expressed as <math>\Delta x</math> of partition <math>i</math>.

<math>= \sum_{i=1}^n [f(c_i)(\Delta x_i)]</math>

Notice that we are describing the area of a rectangle, with the width times the height, and we are adding the areas together.  Each rectangle, by virtue of the [[Mean Value Theorem]], describes an approximation of the curve section it is drawn over.  Also notice that <math>\Delta x_i</math> does not need to be the same for any value of <math>i</math>, or in other words that the width of the rectangles can differ.  What we have to do is approximate the curve with <math>n</math> rectangles.  Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we will get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the [[Riemann integral]].  That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

<math>\lim_{\| \Delta \| \to 0} \sum_{i=1}^n [f(c_i)(\Delta x_i)] \equiv \int_{b}^{a} f(x)\,dx</math>

== Formal statements ==
Stated formally, the theorem says:

<blockquote style="padding: 1em; border: 2px dotted purple;">
If the [[function (mathematics)|function]] ''g''(''x'') is continuous on some [[interval (mathematics)|interval]] [''a'', ''b''], then there exist infinitely many antiderivatives ''G''(''x'') whose derivatives are ''g''(''x'').

If the [[function (mathematics)|function]] ''f' ''(''x'') is continuous on some [[interval (mathematics)|interval]] [''p'', ''q''] and ''f''(''x'') is one of its antiderivatives, then

:<math>\int_a^b f'(x) dx = f(b) - f(a)  = \Delta f(x)</math>

:<math>f(x) = \int_a^x f'(t)\, dt + f(a)</math>

if ''a'', ''b'', and ''x'' are in [''p'', ''q''].

Differentiating both sides, we find:

:<math>f'(x) = \frac{d}{dx} \int_a^x f'(t)\, dt.</math>

</blockquote>

As an example, suppose you need to calculate

:<math>\int_2^5 x^2\, dx </math>
 
Here, <math>f(x) = x^2</math> and we can use <math>F(x) = (1/3) x^3</math> as antiderivative. Therefore:

:<math>\int_2^5 x^2\, dx = F(5) - F(2) = {125 \over 3} - {8 \over 3} = {117 \over 3} = 39.</math>

== Generalizations ==
We don't need to assume continuity of ''f'' on the whole interval. Part I of the theorem then says: if ''f'' is any [[Lebesgue integration|Lebesgue integrable]] function on <math>[a, b]</math> and <math>x_0</math> is a number in <math>[a, b]</math> such that <math>f</math> is continuous at <math>x_0</math>, then 

:<math>F(x) = \int_a^x f(t)\, dt</math>

is differentiable for <math>x = x_0</math> with <math>F'(x_0) = f(x_0)</math>. We can relax the conditions on ''f'' still further and suppose that it is merely locally integrable. In that case, we can conclude that the function ''F'' is differentiable [[almost everywhere]] and ''F'(x)=f(x)'' almost everywhere. This is sometimes known as '''Lebesgue's differentiation theorem'''.

Part II of the theorem is true for any Lebesgue integrable function ''f'' which has an antiderivative ''F'' (not all integrable functions do, though).

The version of [[Taylor's theorem]] which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem.

There is a version of the theorem for [[complex number|complex]] functions: suppose ''U'' is an open set in '''C''' and ''f'': ''U'' <tt>-></tt> '''C''' is a function which has a [[holomorphic function|holomorphic]] antiderivative ''F'' on ''U''. Then for every curve &gamma; : [''a'', ''b''] <tt>-></tt> ''U'', the [[curve integral]] can be computed as

:<math>\oint_{\gamma} f(z) \,dz = F(\gamma(b)) - F(\gamma(a)).</math>

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on [[manifold]]s. 

The most powerful statement in this direction is [[Stokes' theorem]].

== References ==
* Stewart, J. (2003). Fundamental Theorem of Calculus. In Integrals. In ''Calculus: early transcendentals''. Belmont, California: Thomson/Brooks/Cole.

* Larson, Ron,  Bruce H. Edwards, David E. Heyd. ''Calculus of a single variable''. 7th ed.  Boston: Houghton Mifflin Company, 2002.

[[Category:Calculus]]
[[Category:Theorems]]

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