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CAL-22

Midpoint rule

∫_a^b f ≈ (b−a) f((a+b)/2). Rectangle of midpoint height.

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QuadratureMidpoint rule

Governing equation

∫abf≈(b−a) f(a+b2)\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr)

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

where

a
a (—)
b
b (—)
f(m)
f at midpoint (—)
I
Midpoint 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-22 · curve
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①

Governing law — red

∫abf≈(b−a) f(a+b2)\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr)

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

f(m)=−20f(m) = -20I=−40I = -40

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

∫abf≈(b−a) f(a+b2)\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr)

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

f(m)=−20f(m) = -20I=−40I = -40

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′=2I^{\prime} = 2

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-22 — Midpoint rule) is the form associated with Midpoint rule. Working symbols: aa, bb, f(m)f(m) →\rightarrow II. Error −(b−a)³ f″(ξ)/24 — half the trapezoid error, opposite sign.

Purpose

Purpose: compute II from aa, bb, f(m)f(m) in Calculus via ∫abf≈(b−a) f(a+b2)\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr) ∫_a^b f ≈ (b−a) f((a+b)/2). Rectangle of midpoint height. 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(m)=3.000 —f(m) = 3.000\,\mathrm{—}, the governing relation ∫abf≈(b−a) f(a+b2)\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr) yields I=6.0000 —I = 6.0000\,\mathrm{—}. One panel, sample at the midpoint. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Midpoint estimate I6.0000 —
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CAL-22 · curve
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Narration of this film

One panel, sample at the midpoint.

Error −(b−a)³ f″(ξ)/24 — half the trapezoid error, opposite sign.

Reading speed

Watch on YouTube