🧪 Math Lab · Chapter 3

🧮 ∫ Integral Lab 🔬

How do you measure a wiggly curvy shape when there's no formula for it? Chop it into tiny strips and add them up!

Built for Maksim. Everything on this page moves — poke it! 👉

strips1 total area of strips0.00 true area16.80

↓ Start at the easiest shape in the world.

1

📐 Area is just counting squares

Drag the orange dot.
Width
5
Height
3
Squares inside
15

Drag the dot, or use the arrow keys after clicking the picture.

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 peasy. 🍋

2

😵 Now count this one

Same grid. Curvy top. Uh oh.
Definitely inside
—
At most
—

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 way better. 🚀

3

✂️ Chop it into strips

Rectangles are easy. So use rectangles!

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 — boom — you get an estimate!

strips (your guess)the real curve
4
Strips add up to
—
True area
16.80
Off by
—

The bar shows how close you are. Try 1 strip, then 6, then 60!

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. 📉

4

🔁 Chop again. And again. And again.

Press the button until something magic happens.
StripsThey add up toWrong by

Here's the big idea. The strips never quite give the right answer… but they get closer and closer to one exact number, and they never wander off somewhere else. That number they're heading for is the real area. Scientists call this convergence. ✨

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.

5

🔍 The ∫ sign, decoded

Tap each part.

∫ — "add up all of them"

It's a stretched-out letter S, for Sum. Somebody wrote it by hand in 1675 and we never stopped. 🖋️

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.

6

🧪 Your turn: the area playground

Drag the two purple handles to choose where to start and stop.
24
From
1.0
To
5.0
Area (the integral)
—

Try this: drag A and B onto the same spot. Zero width, zero area — the integral says 0 too! (Keyboard: click the picture, then ← → move A, and Shift + arrows move B.)

7

➖ Below the line counts as minus

Drag the handles across the dip.
Above the line
—
Below the line
—
Integral (the total)
—

Yellow strips add. Pink strips take away!

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 pink exactly cancels the yellow and the integral lands on 0.00. 🎯

8

🚲 Why anyone cares: speed → distance

Press play and watch the bike.

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?

0.0 s
Speed right now
3.0 m/s
Distance so far
0.0 m

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! 💡

9

🎯 Guess the area

Five rounds. Every square is worth 1.
Round 1 of 5 Score 0
10.0
Count the squares. Half-squares are allowed!
★

🏆 What you just learned

This is real university math. You did it at 8!
  • Area means counting squares — and rectangles are the only shape that's easy.
  • Any curvy shape can be chopped into thin rectangular strips.
  • More strips, smaller mistake. The sums converge on one exact number.
  • That exact number is the integral, written ∫ f(x) dx.
  • Area under the line is positive; area below it is negative.
  • The area under a speed graph is a distance. Integrals turn one thing into another!

Take it further: open your 2D plotter, draw y = x², and guess the area from 0 to 3 before you check. (The answer is 9!) Then try y = x³ from 0 to 2 — the answer is 4. Spot the pattern yet? 🕵️