CAL-19

Graph arc-length snapshot

L ≈ √(1 + (y')²) Δx. One-panel arc of y = f(x).

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CurvesArc length

Governing equation

\Delta s=\sqrt{1+(y')^2}\,\Delta x

where

y': Slope (—)
\Delta x: Width (—)
\Delta s: Arc panel (—)

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-19 — Graph arc-length snapshot) is the form associated with Arc length. Working symbols: $y'$ , $\Delta x$ $\rightarrow$ $\Delta s$ . ds = √(dx² + dy²) = √(1 + y'²) dx. Composite sums the panels.

Purpose

Purpose: compute

\Delta s

from

y'

\Delta x

in Calculus via

\Delta s=\sqrt{1+(y')^2}\,\Delta x

L ≈ √(1 + (y')²) Δx. One-panel arc of y = f(x). Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

y' = 1.000\,\mathrm{—}

\Delta x = 0.500\,\mathrm{—}

, the governing relation

\Delta s=\sqrt{1+(y')^2}\,\Delta x

yields

\Delta s = 0.7071\,\mathrm{—}

. Enter a slope y' and a width Δx. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Slope y'1 —Width \Delta x0.5 —

Outputs

Arc panel \Delta s0.7071 —

Reading speed

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Narration of this film

Enter a slope y' and a width Δx.

ds = √(dx² + dy²) = √(1 + y'²) dx. Composite sums the panels.

Reading speed

Watch on YouTube