∫ Integrals

Every square in the grid is 1 unit of area. Count them: that's the area. For a rectangle you don't even have to count — width × height does it for you. Easy.

There's no width × height here, because the top keeps changing height. Counting whole squares gives you a floor and a ceiling — the real answer is trapped somewhere in the middle. That's a start, but it's a sloppy answer, and Max, you can do better.

Here's the whole trick. Slice the shape into tall thin strips, like slicing bread. Each strip is nearly a rectangle, and you already know how to do rectangles. Add all the strips together and you get an estimate.

With a few fat strips the answer is bad — strips poke out above the curve or leave gaps below it. Add more strips and the pokes and gaps get tiny. The mistake shrinks faster than the strips do.

Imagine strips so thin that a million of them fit in the shape. Then a billion. Then thinner than you can imagine. The answer they're sneaking up on — that exact number — is called the integral.

Read out loud, ∫₀⁶ f(x) dx is: add up, from 0 to 6, the height times a super thin width. Which is exactly what you were doing with the slider. The symbol is just shorthand for chopping.

When the curve dips under the line, the strip height is negative, so the strip subtracts instead of adding. Think of walking forward and then backward: the backward steps take away from how far you got. Find a spot where the red exactly cancels the yellow and the integral lands on 0.00.

You're on your bike. You speed up, you slow down, you speed up again. Nobody measured how far you rode — but a sensor recorded your speed the whole time. Can you work out the distance?

Speed × time = distance, but only if the speed never changes. It changes constantly here — so chop the ride into slices of time so short that the speed barely moves inside each one. Distance in one slice ≈ speed × tiny bit of time. Add them all up: ∫ speed dt. The shaded area under a speed graph is distance travelled. Same trick, brand new meaning.