Single-variable calculus from first principles
In the last two lessons you added up a list of numbers and asked where the running total was heading. Now we make one bold leap: what if the things we're adding are infinitely many, infinitely thin pieces? That single move — sum tiny pieces, then take a limit — is the whole idea of the integral.
Here's the picture. You want the area under a curve, but the top is wavy, so there's no single height to multiply by the width. So you cheat, carefully: cover the region with thin vertical rectangles, each so narrow the curve is almost flat across it. Add up their areas. You won't get the exact answer — the rectangle tops poke above or fall below the curve — but you'll get close. Then make the rectangles thinner.
To find the area of an oddly-shaped region, imagine filling it with many thin vertical strips, like stacking a row of coins side by side under the curve. Each strip is so narrow that its top is almost flat, so you can treat it as a simple rectangle and add up the areas. The thinner you slice the strips — the smaller you make Δx — the more snugly the stack fills the region, and the area you get closes in on the exact answer.