Integrals — Measuring a Shape With No Formula
What is an integral?
chop a curved shape into thinner and thinner strips to measure its area
what this teaches: integrals, area under a curve, riemann sums
Play →What you'll learn
- How to estimate the area under a curve by counting squares, and why the estimate stays fuzzy
- What a Riemann sum is, and how left, middle, and right strips give different answers
- Why thinner and thinner strips close in on one exact area
- What each part of the integral notation actually stands for
- Why the area under a speed graph tells you the distance traveled
How to play
- Start by counting squares under the curve, first the ones fully inside and then the ones it touches, to trap the answer between two numbers.
- Switch to strips and choose whether to measure from the left edge, the middle, or the right edge.
- Press Double the strips over and over and watch the three estimates close in on each other.
- Try the area playground and the guess-the-area round to test yourself.
FAQ
What age is this for?
Around age 14 and up. It starts from counting squares, so the entry point is much earlier than the topic usually suggests.
Should I do derivatives first?
It helps, but it is not required. This page builds the area idea from scratch and stands on its own; the link back to derivatives shows up at the end.
Do I need to install anything?
No. It runs in any web browser, it is free, and there is no signup or account.
For parents and teachers
The squeeze between the fully-inside count and the touching count is the heart of this, and it is worth pausing on. It shows the exact area trapped between two numbers that draw closer as the strips get thinner, which is the limit concept doing real work several steps before anyone writes down its formal definition.