Cover It in Rectangles, Then Ask Which Way They Lean

You'll learnto build a Riemann sum with the right width, the right sample points and the right number of terms, and to decide whether an approximation comes out high or low from the feature that actually settles it — direction for rectangles, curvature for trapezoids — instead of from the name of the rule.

There is no formula for the area under a curved graph, so we cover it with shapes that do have one. Two things then go wrong, and they go wrong independently. The setup uses the wrong width, the wrong sample points, or one rectangle too many. And the direction of the error gets memorised as a fact about the word left, when it is a fact about the function — and half the time about a different feature of the function than the one being looked at.

8 STEPS · 6 QUICK CHECKS · THE PICTURE COMPUTES EVERY VERDICT, NOTHING IS ASSERTED · v1

Δx = (b − a)/n · n strips, n + 1 grid points · direction decides rectangles, curvature decides trapezoids

Before you start

What you're looking at

One curve, with the exact area under it shaded in blue. Rectangles and trapezoids are laid over that shading, so wherever a shape sticks out past the blue it has added area that is not there, and wherever the blue shows above a shape it has missed some.

The question

Does an approximation come out too high or too low — and which feature of the function decides it?

Watch for

The colours are computed, never chosen. Each piece is compared with the exact area over its own strip: pink means that piece falls short, violet means it overshoots. When the same rule changes colour between two steps, nothing about the rule changed — the function did.

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