Definition:Improper Integral on Open Above Interval/Mistake

Source Work

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.):

infinite integral (improper integral)

Mistake

An integral ... whose integrand is a function $\map {\mathrm f} x$ that is finite for $a \le x < b$, but infinite for $x = b$, is

$\ds \int \limits_a^b \map {\mathrm f} x \rd x$

which is short for

$\ds \lim_{\delta \mathop \to \infty} \int \limits_a^{b - \delta} \map {\mathrm f} x \rd x$

where $\delta > 0$.

Correction

The $\lim$ expression is incorrect.

It should read:

$\ds \lim_{\delta \mathop \to 0} \int \limits_a^{b - \delta} \map {\mathrm f} x \rd x$

This is correct in the $2$nd edition.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinite integral (improper integral)