CAL-06

Trapezoid-rule snapshot

∫_a^b f ≈ (b−a)(f(a)+f(b))/2. One panel of the composite rule.

Reading speed

QuadratureTrapezoid rule

Governing equation

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

$b \in \left[-5, 8\right]$

where

a: a (—)
b: b (—)
f(a): f(a) (—)
f(b): f(b) (—)
I: Integral 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.

00:0 / 00:08

①

Governing law — red

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

$b \in \left[-5, 8\right]$

$b = -5$ $I = -12.5$

Along the curve

②

Domain of the variable — blue

b_{\min} = -5,\quad b_{\max} = 8

$b \in \left[-5, 8\right]$

$b = -5$ $b = -5$

Along the curve

③

Response curve — green

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

$b \in \left[-5, 8\right]$

$b = -5$ $I = -12.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}b}

$b \in \left[-5, 8\right]$

$b = -4.918$ $I^{\prime} = 2.5$

Along the curve

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-06 — Trapezoid-rule snapshot) is the form associated with Trapezoid rule. Working symbols: $a$ , $b$ , $f(a)$ , $f(b)$ $\rightarrow$ $I$ . Error −(b−a)³ f″(ξ)/12. Composite: sum of panels, error O(h²).

Purpose

Purpose: compute

I

from

a

b

f(a)

f(b)

in Calculus via

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

∫_a^b f ≈ (b−a)(f(a)+f(b))/2. One panel of the composite rule. 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 = 1.000\,\mathrm{—}

f(a) = 1.000\,\mathrm{—}

f(b) = 4.000\,\mathrm{—}

, the governing relation

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

yields

I = 2.5000\,\mathrm{—}

. A curve, a trapezoid under it. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a0 —b b1 —f(a) f(a)1 —f(b) f(b)4 —

Outputs

Integral estimate I2.5000 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A curve, a trapezoid under it.

Error −(b−a)³ f″(ξ)/12. Composite: sum of panels, error O(h²).

Reading speed

Watch on YouTube