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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

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

y′∈[−8,8]y' \in \left[-8, 8\right]

where

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

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-19 · curve
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①

Governing law — red

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

y′∈[−8,8]y' \in \left[-8, 8\right]

y′=−8y' = -8Δs=4.031\Delta s = 4.031

②

Domain of the variable — blue

ymin⁡′=−8,ymax⁡′=8y'_{\min} = -8,\quad y'_{\max} = 8

y′∈[−8,8]y' \in \left[-8, 8\right]

y′=−8y' = -8y′=−8y' = -8

③

Response curve — green

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

y′∈[−8,8]y' \in \left[-8, 8\right]

y′=−8y' = -8Δs=4.031\Delta s = 4.031

④

Rate of change — amber

dΔsdy′\frac{\mathrm{d}\Delta s}{\mathrm{d}y'}

y′∈[−8,8]y' \in \left[-8, 8\right]

y′=−7.899y' = -7.899Δs′=−0.496\Delta s^{\prime} = -0.496

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′y', Δx\Delta x →\rightarrow Δs\Delta s. ds = √(dx² + dy²) = √(1 + y'²) dx. Composite sums the panels.

Purpose

Purpose: compute Δs\Delta s from y′y', Δx\Delta x in Calculus via Δs=1+(y′)2 Δx\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 —y' = 1.000\,\mathrm{—}, Δx=0.500 —\Delta x = 0.500\,\mathrm{—}, the governing relation Δs=1+(y′)2 Δx\Delta s=\sqrt{1+(y')^2}\,\Delta x yields Δs=0.7071 —\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

Outputs

  • Arc panel \Delta s0.7071 —
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CAL-19 · curve
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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.

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