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

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

where

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

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: $a$ , $b$ , $f(a)$ , $f(m)$ , $f(b)$ $\rightarrow$ $I$ . Exact for cubics. Composite error O(h⁴).

Purpose

Purpose: compute

I

from

a

b

f(a)

f(m)

f(b)

in Calculus via

\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\,\mathrm{—}

b = 2.000\,\mathrm{—}

f(a) = 1.000\,\mathrm{—}

f(m) = 2.000\,\mathrm{—}

f(b) = 1.000\,\mathrm{—}

, the governing relation

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

yields

I = 3.3333\,\mathrm{—}

. One Simpson panel, three samples. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a0 —b b2 —f(a) f(a)1 —f(mid) f(m)2 —f(b) f(b)1 —

Outputs

Simpson estimate I3.3333 —

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Narration of this film

One Simpson panel, three samples.

Exact for cubics. Composite error O(h⁴).

Reading speed

Watch on YouTube