Fundamental Theorem of Calculus/Motivation

Motivation for Fundamental Theorem of Calculus

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:

• the error function:

$\map {\erf} x = \ds \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t$

• the sine integral function

$\map \Si x = \ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$

Sources

• 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $2$ Fundamental Theorem of the Calculus