Animation · Trapezoids - COMP 521: Introduction to Scientific Computing and Data Science

Join the two endpoint samples with a straight line and integrate that line. The film adds trapezoidal areas for $e^x$ on $[0,1]$ and compares successive uniform refinements.

Before watching¶

The curve is convex. Predict whether its endpoint chord lies above or below the curve, and what that implies for the integral estimate.

Join the two endpoint samples with a straight line and integrate that line. The film adds trapezoidal areas for e^x on [0,1] and compares successive uniform refinements.

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Mathematical connection¶

Let $I=\int_a^{a+h} f(x)\,dx$ be the exact integral over one panel of width $h$ . The trapezoid rule is

T=\frac{h}{2}[f(a)+f(a+h)].

(1)

Taylor expansion of the endpoint value and integration of the same series give

T=hf(a)+\frac{h^2}{2}f'(a)+\frac{h^3}{4}f''(a)+O(h^4),

(2)

I=hf(a)+\frac{h^2}{2}f'(a)+\frac{h^3}{6}f''(a)+O(h^4).

(3)

Thus $T-I=\frac{h^3}{12}f''(a)+O(h^4)$ . The local error is $O(h^3)$ and the composite error over a fixed interval is $O(h^2)$ for a sufficiently smooth function. The straight-line model integrates constants and linear functions exactly.

Explain what you saw¶

Compare the actual errors after each halving of the panel width. Explain why a ratio near four is expected once the leading second-order composite error dominates. This ratio is an asymptotic prediction, not an unconditional guarantee.

Use the player controls to pause, replay, or view fullscreen. The animation has no narration; its labels and the explanation above carry the content.

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