prop_imp_ints.pdf - Proper and Improper Integrals Definite integrals are defined as limits of Riemann sums(Stewart \u00a75.2 Sometimes the limit process

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

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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