Riemann Sums
The derivative shrank a gap to zero to get a slope. The integral plays the same trick in reverse: chop an area into slices, and let the slices get infinitely thin. For Grade 12.
Slice, add, repeat
The area under a curvy graph looks impossible to measure — curves don't fit formulas for rectangles. So don't measure it. Approximate it: cut the stretch from a to b into n strips, stand a rectangle on each, and add them up:
Every rectangle is either a little too tall or a little too short — but the more slices you use, the smaller the total mistake. As n → ∞ the sum stops being an approximation and becomes the integral, written ∫ f(x) dx.
Crank up the slices 🟠
Drag the blue dots a and b to choose the region; slide n to add slices, or hit "n → ∞". Try each sample-point rule and watch which one homes in fastest.
What to watch
The sum is measurable
For any finite n, the orange number is just rectangle areas added up — width × height, Grade 6 stuff. No limits needed yet.
The integral is the limit
As n → ∞, the orange sum closes in on the green number — the exact area. Same "arbitrarily close" idea as the derivative, pointed at area instead of slope.
Compare the rules on x²: Left under-shoots (the curve rises away from each rectangle), Right over-shoots, and Midpoint nearly cancels its own errors — watch how few slices it needs. Then drag a below zero on sin x: slices under the axis count as negative area (purple), and a symmetric region sums to zero.