Another common reason is that you have a discontinuity (a hole in the graph). What could cause you to not know the interval length? One reason is infinity as a limit of integration. If you can’t divide the interval, you have an improper integral. If you don’t know the length of the interval, then you can’t divide the interval into n equal pieces. However, if your interval is infinite (because of infinity being one if the interval ends or because of a discontinuity in the interval) then you start to run into problems. You’re taking a known length (for example from x = 0 to x = 20) and dividing that interval into a certain amount of tiny rectangles with a known base length (even if it’s an insignificantly tiny length). When you integrate, you are technically evaluating using rectangles with an equal base length (which is very similar to using Riemann sums).
Well-defined, finite upper and lower limits but that go to infinity at some point in the interval:
Limits for improper integrals do not always exist An improper integral is said to converge (settle on a certain number as a limit) if the limit exists and diverge (fail to settle on a number) if it doesn’t. The workaround is to turn the improper integral into a proper one and then integrate by turning the integral into a limit problem. And if your interval length is infinity, there’s no way to determine that interval. one without infinity) is that in order to integrate, you need to know the interval length. The reason you can’t solve these integrals without first turning them into a proper integral (i.e. Improper integrals are integrals you can’t immediately solve because of the infinite limit(s) or vertical asymptote in the interval. Either one of its limits are infinity, or the integrand (that function inside the interval, usually represented by f(x)) goes to infinity in the integral. An improper integral is a definite integral-one with upper and lower limits-that goes to infinity in one direction or another.
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