Animate Riemann sums by filling the area under a curve with rectangles, then increasing their count so the staircase converges to the exact integral. Watching the rectangles thin out is the definition of integration.

The core idea

A definite integral is the area under a curve. A Riemann sum approximates that area with rectangles:

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x$

As the number of rectangles n → ∞, the approximation becomes exact.

The animation, beat by beat

1. Plot the curve f(x) = x² on [0, 2].
2. Add 4 rectangles — a chunky staircase that misses area.
3. Increase n: 4 → 10 → 50 → 200. Rectangles thin; the staircase hugs the curve.
4. Show the running total approaching 8/3 ≈ 2.667.
5. Declare the limit = the integral.

Compare the three rules

| Rule | Rectangle height from | For increasing f | |---|---|---| | Left | left endpoint | underestimates | | Right | right endpoint | overestimates | | Midpoint | midpoint | closest |

Animating all three converging side by side is a memorable lesson. Pairs naturally with Animate the Derivative — the inverse operation.

Manim building blocks

axes.plot for the curve and axes.get_riemann_rectangles — a built-in that generates the rectangles for a given n. Animate Transform between successive n values; a DecimalNumber shows the running sum.

The prompt

"Show the area under f(x)=x² on [0,2] with Riemann rectangles, increasing n from 4 to 200 so the sum converges to 8/3."

→ Render it at quantumsketch.app.

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Written by Shihab Shahriar Antor · Shahriar Labs