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.

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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE PICTURE COMPUTES EVERY VERDICT, NOTHING IS ASSERTED · v2

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. Wherever a shape sticks out past the blue it has added area that is not there; wherever 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? The colors are computed, never chosen: pink falls short, violet overshoots.

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