prop_imp_ints.pdf - Proper and Improper Integrals Definite...

Proper and Improper Integrals
Definite integrals are defined as limits of Riemann sums (Stewart § 5.2).
Sometimes the limit process breaks down and integral expressions don’t ac-
tually mean anything. Fortunately there is a theorem that shows this is not
a problem for a big class of integrals.
Definition : an integral expression
b
a
f ( t ) dt is proper if the limits and func-
tion are bounded . More precisely,
• the limits of integration are finite (i.e. not ±∞ );
• the function is defined and bounded on the interval [ a, b ]; and
• the function has only finitely many discontinuities in the interval.
Theorem Proper integrals are always well-defined (i.e. the limit converges).
An integral is