Fundamental Theorem of Calculus

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Theorem

First Part

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $F$ be a real function which is defined on $\left[{a \,.\,.\, b}\right]$ by:

$\displaystyle F \left({x}\right) = \int_a^x f \left({t}\right) \ \mathrm d t$


Then $F$ is a primitive of $f$ on $\left[{a \,.\,.\, b}\right]$.


Second Part

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Then:

  • $f$ has a primitive on $\left[{a \,.\,.\, b}\right]$
  • If $F$ is any primitive of $f$ on $\left[{a \,.\,.\, b}\right]$, then:
$\displaystyle \int_a^b f \left({t}\right) \ \mathrm d t = F \left({b}\right) - F \left({a}\right) = \left[{ F \left({t}\right) }\right]_a^b$


Notes

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

What it in fact tells us is that, in general, in order 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.