= Fundamental theorem of calculus
{title2=$\frac{d}{dx}\int_a^x f(t)\,dt=f(x)$}
{wiki}
The fundamental theorem of calculus says that \b[accumulation and local change undo one another], under appropriate regularity conditions. If $f$ is a <continuous function> on $[a,b]$ and
$$
A(x)=\int_a^x f(t)\,dt,
$$
then $A$ is <differentiable> for $a<x<b$ and $A'(x)=f(x)$. Conversely, if $G$ is an <antiderivative> of $f$ on this <interval>, continuous at its endpoints, then
$$
\boxed{\int_a^b f(x)\,dx=G(b)-G(a).}
$$
The first statement turns an <integral> into a <function> with a known <derivative>; the second turns an entire accumulation into two endpoint values.
For $f(t)=1+2t$, accumulation from zero gives $A(x)=x+x^2$. At $x=2$, its <derivative> is $A'(2)=5$, exactly the current value $f(2)$. The accumulated value there is $A(2)=6$, and by $x=3$ it has reached $A(3)=12$. The accumulated quantity and its rate are different <functions>, linked by <differentiation>.
\Image[/fundamental-theorem-calculus.png]
{title=The instantaneous rate determines the tangent to the accumulation curve}
This connection makes the theorem useful far beyond geometric <area>. It recovers <displacement> from <velocity>, gives a <probability> from a <probability density function>, and converts many <differential equations> into <integral equations>. The striking economy is that information spread across a whole <interval> can sometimes be recovered from an <antiderivative> at its two boundaries.
A short explanation of the first part is that
$$
\frac{A(x+h)-A(x)}h=\frac1h\int_x^{x+h}f(t)\,dt
$$
is the local average of $f$. <Continuity> makes this average approach $f(x)$. For the second part, $G-A$ has zero <derivative> and is constant by the <mean value theorem>. These are the ideas behind the proof; the endpoint and regularity conditions make them valid.
An early geometric form of this connection appears in Isaac Barrow's \i[Lectiones geometricae] (1670), Lecture X, Proposition 11. The original volume is catalogued by CESIMA, and Michael Nauenberg's historical research reproduces and examines the geometric argument.https://cesimadigital.pucsp.br/handle/bcd/3675[Isaac Barrow, Lectiones geometricae, 1670]{ref}https://arxiv.org/abs/1111.6145[Michael Nauenberg, Barrow and Leibniz on the fundamental theorem of the calculus, 2011]{ref}
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