CAL-25

Arc length snapshot

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Calculusarc length

Governing equation

L=\\int\\sqrt{1+y'^2}\\,dx\\approx\\Delta x\\sqrt{1+m^2}

where

\Delta x: Δx (—)
y': y' (—)
L: L (—)

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-25 — Arc length snapshot) is the form associated with arc length. Working symbols: $\Delta x$ , $y'$ $\rightarrow$ $L$ . One step.

Purpose

Purpose: compute

L

from

\Delta x

y'

in Calculus via

L=\\int\\sqrt{1+y'^2}\\,dx\\approx\\Delta x\\sqrt{1+m^2}

One step. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta x = 1.000\,\mathrm{—}

y' = 0.500\,\mathrm{—}

, the governing relation

L=\\int\\sqrt{1+y'^2}\\,dx\\approx\\Delta x\\sqrt{1+m^2}

yields

L = 1.1180\,\mathrm{—}

. One step. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Δx \Delta x1 —y' y'0.5 —

Outputs

L L1.1180 —

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