Mathematics
Improper Integrals with Infinite Bounds and Discontinuities
Quick fact
The integral from 0 to infinity of e^{-x} dx equals exactly 1, even though it sums an infinite area. This is because the function decays so quickly that the total accumulated area is finite.
Why this is interesting
You've been told that the area under a curve can be finite even when the curve stretches to infinity. How can something infinite have a finite area?