CAL-05

Fundamental theorem snapshot

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

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IntegralsFTC

Governing equation

\int_a^b F'=F(b)-F(a)

where

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

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: $F(b)$ , $F(a)$ $\rightarrow$ $\int$ . The derivative of the accumulation function is the integrand. Evaluation is a difference of the antiderivative.

Purpose

Purpose: compute

\int

from

F(b)

F(a)

in Calculus via

\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\,\mathrm{—}

F(a) = 1.000\,\mathrm{—}

, the governing relation

\int_a^b F'=F(b)-F(a)

yields

\int = 4.0000\,\mathrm{—}

. Enter F at the two endpoints. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

F(b) F(b)5 —F(a) F(a)1 —

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.

Reading speed

Watch on YouTube