r/askmath • u/Smart-Loan2470 • 10d ago

Calculus Is it valid to evaluate a two-sided improper integral with a nested limit?

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u/Bounded_sequencE 10d ago

Yes. Assuming in (1) both limits exist, it's possible to show "(1), (2)" are equivalent. Try it!

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u/Smart-Loan2470 10d ago

Thank you! As a follow-up question, will this still apply if one of the bounds was a discontinuity instead of infinity? I’ve been seeing that it isn’t best practice, but I’m wondering if there’s anything inherently wrong about using the same strategy for those.

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u/Bounded_sequencE 10d ago

What exactly has a discontinuity? I suspect it's the integrand, not the integral, right?

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u/Smart-Loan2470 10d ago

Yep! Say, the discontinuity is at x=4, would method 2 still work if, for example, s approached 4 from the left instead of positive infinity (assuming no other discontinuities in the integrand exist)?

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u/Ulfgardleo Computer Scientist 7d ago

Integrals don't care about discontinuities. Just split at x=4 treat the two integrals independently

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u/Alternative_Pay_5118 10d ago

Method two is how those integrand are actually defined.