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CAL-05

Fundamental theorem snapshot

∫_a^b F' = F(b) − F(a). Net change of an antiderivative.

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IntegralsFTC

Governing equation

∫abF′=F(b)−F(a)\int_a^b F'=F(b)-F(a)

F(b)∈[−50,50]F(b) \in \left[-50, 50\right]

where

F(b)
F(b) (—)
F(a)
F(a) (—)
\int
Definite integral (—)

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.

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①

Governing law — red

∫abF′=F(b)−F(a)\int_a^b F'=F(b)-F(a)

F(b)∈[−50,50]F(b) \in \left[-50, 50\right]

F(b)=−50F(b) = -50∫=−51\int = -51

②

Domain of the variable — blue

F(b)min⁡=−50,F(b)max⁡=50F(b)_{\min} = -50,\quad F(b)_{\max} = 50

F(b)∈[−50,50]F(b) \in \left[-50, 50\right]

F(b)=−50F(b) = -50F(b)=−50F(b) = -50

③

Response curve — green

∫abF′=F(b)−F(a)\int_a^b F'=F(b)-F(a)

F(b)∈[−50,50]F(b) \in \left[-50, 50\right]

F(b)=−50F(b) = -50∫=−51\int = -51

④

Rate of change — amber

d∫dF(b)\frac{\mathrm{d}\int}{\mathrm{d}F(b)}

F(b)∈[−50,50]F(b) \in \left[-50, 50\right]

F(b)=−49.37F(b) = -49.37∫′=1\int^{\prime} = 1

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-05 — Fundamental theorem snapshot) is the form associated with FTC. Working symbols: F(b)F(b), F(a)F(a) →\rightarrow ∫\int. The derivative of the accumulation function is the integrand. Evaluation is a difference of the antiderivative.

Purpose

Purpose: compute ∫\int from F(b)F(b), F(a)F(a) in Calculus via ∫abF′=F(b)−F(a)\int_a^b F'=F(b)-F(a) ∫_a^b F' = F(b) − F(a). Net change of an antiderivative. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given F(b)=5.000 —F(b) = 5.000\,\mathrm{—}, F(a)=1.000 —F(a) = 1.000\,\mathrm{—}, the governing relation ∫abF′=F(b)−F(a)\int_a^b F'=F(b)-F(a) yields ∫=4.0000 —\int = 4.0000\,\mathrm{—}. Enter F at the two endpoints. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Definite integral \int4.0000 —
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Narration of this film

Enter F at the two endpoints.

The derivative of the accumulation function is the integrand. Evaluation is a difference of the antiderivative.

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