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

\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr)

where

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

Purpose

Purpose: compute

I

from

a

b

f(m)

in Calculus via

\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\,\mathrm{—}

b = 2.000\,\mathrm{—}

f(m) = 3.000\,\mathrm{—}

, the governing relation

\int_a^b f\approx(b-a)\,f\bigl(\tfrac{a+b}2\bigr)

yields

I = 6.0000\,\mathrm{—}

. One panel, sample at the midpoint. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a0 —b b2 —f at midpoint f(m)3 —

Outputs

Midpoint estimate I6.0000 —

Reading speed

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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