Fundamental Theorem of Calculus

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First Part

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Let $F$ be a real function which is defined on $\closedint a b$ by:

$\ds \map F x = \int_a^x \map f t \rd t$

Then $F$ is a primitive of $f$ on $\closedint a b$.

Second Part

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.


$(1): \quad f$ has a primitive on $\closedint a b$
$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then:
$\ds \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$


It can be seen that, to all intents and purposes, the first part and the second part of the Fundamental Theorem of Calculus are converses of each other.

What it tells us is that, in general, to work out the value of a definite integral, we do not have to flog through the difficult and tedious work of calculating it from first principles.

All we need to do is work out the formula for the antiderivative.

This of course can only be done if the function in question does in fact have an antiderivative.

In cases where it does not, or it can not be calculated, then it may well be necessary to go back to first principles after all.

However, what it does allow us to do is to define such functions as definite integrals, for example:

$\map {\erf} x = \ds \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t$
$\map \Si x = \ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$

Historical Note

In $1668$, James Gregory published Geometriae Pars Universalis, in which the Fundamental Theorem of Calculus first makes its appearance, although only for a limited class of functions.

It is believed that the earliest complete statement and proof was made by Isaac Newton.

This can be seen in a letter to Leibniz from $1676$ or $1677$, collected as item $190$ of 1959 -- 1961: H.W. Turnbull: The Correspondence of Isaac Newton.

Isaac Barrow is also cited by some as being the first to establish it.

Leibniz himself, in his own turn, claimed to have made the same startling realisation on reading Blaise Pascal's $1658$ work Traité des Sinus du Quart de Cercle.

In Leibniz's $1684$ article Nova Methodus pro Maximis et Minimis, published in Acta Eruditorum, he takes this result as given, stating that $\int$ and $\d$ are each other's converse, with no attempt at proof.