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CAL-07

Simpson's rule snapshot

∫ ≈ (b−a)/6 (f(a)+4f(m)+f(b)), m midpoint. Parabolic panel.

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QuadratureSimpson

Governing equation

∫≈b−a6(f(a)+4f(m)+f(b))\int\approx\dfrac{b-a}{6}\bigl(f(a)+4f(m)+f(b)\bigr)

f(m)∈[−20,20]f(m) \in \left[-20, 20\right]

where

a
a (—)
b
b (—)
f(a)
f(a) (—)
f(m)
f(mid) (—)
f(b)
f(b) (—)
I
Simpson estimate (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

CAL-07 · curve
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①

Governing law — red

∫≈b−a6(f(a)+4f(m)+f(b))\int\approx\dfrac{b-a}{6}\bigl(f(a)+4f(m)+f(b)\bigr)

f(m)∈[−20,20]f(m) \in \left[-20, 20\right]

f(m)=−20f(m) = -20I=−26I = -26

②

Domain of the variable — blue

f(m)min⁡=−20,f(m)max⁡=20f(m)_{\min} = -20,\quad f(m)_{\max} = 20

f(m)∈[−20,20]f(m) \in \left[-20, 20\right]

f(m)=−20f(m) = -20f(m)=−20f(m) = -20

③

Response curve — green

∫≈b−a6(f(a)+4f(m)+f(b))\int\approx\dfrac{b-a}{6}\bigl(f(a)+4f(m)+f(b)\bigr)

f(m)∈[−20,20]f(m) \in \left[-20, 20\right]

f(m)=−20f(m) = -20I=−26I = -26

④

Rate of change — amber

dIdf(m)\frac{\mathrm{d}I}{\mathrm{d}f(m)}

f(m)∈[−20,20]f(m) \in \left[-20, 20\right]

f(m)=−19.75f(m) = -19.75I′=1.333I^{\prime} = 1.333

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-07 — Simpson's rule snapshot) is the form associated with Simpson. Working symbols: aa, bb, f(a)f(a), f(m)f(m), f(b)f(b) →\rightarrow II. Exact for cubics. Composite error O(h⁴).

Purpose

Purpose: compute II from aa, bb, f(a)f(a), f(m)f(m), f(b)f(b) in Calculus via ∫≈b−a6(f(a)+4f(m)+f(b))\int\approx\dfrac{b-a}{6}\bigl(f(a)+4f(m)+f(b)\bigr) ∫ ≈ (b−a)/6 (f(a)+4f(m)+f(b)), m midpoint. Parabolic panel. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given a=0.000 —a = 0.000\,\mathrm{—}, b=2.000 —b = 2.000\,\mathrm{—}, f(a)=1.000 —f(a) = 1.000\,\mathrm{—}, f(m)=2.000 —f(m) = 2.000\,\mathrm{—}, f(b)=1.000 —f(b) = 1.000\,\mathrm{—}, the governing relation ∫≈b−a6(f(a)+4f(m)+f(b))\int\approx\dfrac{b-a}{6}\bigl(f(a)+4f(m)+f(b)\bigr) yields I=3.3333 —I = 3.3333\,\mathrm{—}. One Simpson panel, three samples. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Simpson estimate I3.3333 —
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CAL-07 · curve
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Narration of this film

One Simpson panel, three samples.

Exact for cubics. Composite error O(h⁴).

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Watch on YouTube