Definition:Improper Integral on Open Above Interval/Mistake
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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)