Improper integral
An integral with an infinite limit, or a function that blows up inside the interval. Replace the problem point with a variable and take a limit.
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Analogy
A road that goes on forever: you ask whether the total distance walked settles to a finite number.
Example
Integral from 1 to R is 1 - 1/R.As R tends to infinity, 1/R tends to 0.Limit: 1.(1/x from 1 to infinity diverges.)
In practice
Used in probability, physics and for deciding whether a series converges.
Practice
Flashcards
- What is the integral of 1/x^2 from 1 to infinity?1.
Interview questions
- Does the integral of 1/x^2 from 1 to infinity converge? Compare with 1/x.
- Replace infinity with R and take a limit
- Integral of 1/x^2 from 1 to R is 1 - 1/R
- Limit is 1, so it converges
- Integral of 1/x gives ln R, which diverges
Questions people ask
How can an infinite area be finite?
The curve gets close to the x-axis fast enough that the thin tail adds up to a small amount. For 1 over x squared on 1 to infinity, the area is exactly 1. The slower 1 over x does not shrink fast enough, so its area is infinite.
What does it mean for an improper integral to diverge?
The limit either does not exist or is infinite, so the area has no finite value. For 1 over x from 1 to R, the answer is ln R, which grows without bound as R grows. We then say the integral diverges.