Improper Integrals

Calculus 2 · Axiom Academy

What an integral means when the interval never ends, or the function blows up inside it. 1. When the Interval Never Ends An interval like has no right endpoint, so there is nothing to substitute into an antiderivative — is not a number you evaluate at. Instead, stop at a finite b , do the ordinary definite integral you already know how to do, and then let b run. A finite limit, so this converges — and its value is exactly 1 . No finite limit, so this diverges . There is no value to report. The other way an integral turns improper: the interval is perfectly finite, but the integrand is unbounded somewhere on it. On [0,1] the function has a vertical asymptote at the left endpoint — x = 0 is not even in its domain, so once again there is nothing to substitute. The repair is the same shape: stop short at t , then slide t in toward the trouble. Type 2 — an infinite discontinuity Integrate from t to the right end and take . Integrate from the left end to t and take . Split the interval at the bad point c and handle each side separately. Both halves must converge. The Fundamental Theorem still hands you a number — a meaningless one. Scan the interval first. The height of the strip at t grows without any bound at all — it leaves the top of the frame and never comes back. The area it sweeps out does not. That is the whole surprise of Type 2: an unbounded region can still have a finite area, provided the blow-up is gentle enough. The trap: evaluating straight through a discontinuity

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