Infinite Areas

When infinity adds up to something finite

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Improper integrals are limits: of the upper bound going to ∞, or of a bound approaching a spike.
• $\int_1^\infty x^{-p}\,dx$ is finite exactly when $p > 1$.
• Gabriel’s horn: finite volume, infinite surface area.

Steps

1. Endless, but finite?
2. Two tails
3. An infinite integral
4. The dividing line
5. Gabriel’s horn
6. Infinitely tall, finite area