Foundations of Integral Calculus

date2026-07-15document_iddoc_5fce53fbba2870154685477331b36f69description積分法を、定積分の定義、不定積分、基本定理、計算法へ進むための全体像として整理する講義である。prerequisites極限と連続 / 導関数の定義と差商type講義content_typelecturestatusactive

mathcalculusintegrallecture

1Introduction

This lecture explains how a definite integral is defined as an accumulation of local contributions and how an indefinite integral describes the family of antiderivatives. It also introduces the fundamental theorem of calculus as the connection between these two concepts.

2Learning Sequence

Define the definite integral through Riemann sums.
Interpret a definite integral as signed area and distinguish it from geometric area, which is always nonnegative.
On an interval $I$ , distinguish one antiderivative from the indefinite integral obtained by adding an arbitrary constant to it.
Use the fundamental theorem of calculus to connect accumulation with differentiation.
Compute integrals through substitution and integration by parts.

3Roles of the Two Types of Integral

Concept	Input	Output	Principal interpretation
Definite integral $\int_{a}^{b} f (x) d x$	A function $f$ and an interval $[a, b]$	A single number	Signed accumulation over an interval
Indefinite integral $\int f (x) d x$	A function $f$ on an interval $I$	The family ${F + C ∣ C \in R}$ satisfying $F^{'} = f$	All functions whose derivative is $f$

Definite and indefinite integrals use the same integral symbol, but they have different definitions and different kinds of output. Under suitable hypotheses, the fundamental theorem of calculus guarantees that a definite integral can be evaluated through an antiderivative.

4Related Material

data/lecture/math/calculus/integral-definition-riemann-sums-and-signed-area.lecture.n.md data/lecture/math/calculus/antiderivatives-and-indefinite-integrals.lecture.n.md data/lecture/math/calculus/fundamental-theorem-of-calculus.lecture.n.md data/lecture/math/calculus/integration-rules-and-computation.lecture.n.md

5Exercises

data/exercise/math/calculus/riemann-sums-and-definite-integrals.exercise.n.md data/exercise/math/calculus/antiderivatives-and-indefinite-integrals.exercise.n.md data/exercise/math/calculus/fundamental-theorem-of-calculus.exercise.n.md data/exercise/math/calculus/substitution-and-integration-by-parts.exercise.n.md