Question

What are the two general ways in which an improper integral may occur?

1) By having an unbounded interval of integration
2) By having a function that is not defined at one or more points within the interval of integration
3) By having a function that is not integrable over the interval of integration
4) By having a function that is not continuous over the interval of integration

Answer 1

Improper integrals occur either because the interval of integration is unbounded or because the function is not continuous within the integration interval, which may involve points where the function is undefined.

Step-by-step explanation:

There are two general situations in which an improper integral may arise:

These situations lead to improper integrals because the areas under the curve can potentially be infinite or indefinite. For the first case, the integration might extend to infinity, such as ∞ or ∞. In the second case, there might be a singularity or point of discontinuity within the interval, like a division by zero, that makes the function undefined at that point. Both require special techniques to evaluate, such as taking limits to deal with the infinite bounds or singularities.