A Study on the Concept of Integral through Various Integrals

Introduction to the Study of Integral Concepts

Rationale and Educational Context

7th Mathematics Curriculum Changes: Under the current 7th National Mathematics Curriculum in South Korea, Calculus has been reduced in scale (Mathematics II, Calculus) or deleted (Mathematics I). This reduction conflicts with the goal of fostering "Mathematical Power," which includes problem-solving, creative, logical, and critical thinking, reasoning, communication, and confidence.
Historical Precedents for School Calculus: Early 20th-century pioneers like John Perry (1850–1920) and Felix Klein (1849–1925) led the modernization of mathematics education. Perry argued that calculus uncovers the mysteries of nature and develops thinking ability, while Klein believed calculus was essential for deep understanding of functions.
Practical Utility: Calculus is vital in various fields, including measuring vehicle speed, analyzing stock trends, the basic principles of CT scans, satellite technology, computer rocket launches, weather forecasting, and 3D graphics in games.
Goals of the Thesis:
1. To examine the historical development of calculus (focusing on integration) from ancient Greece to the modern era.
2. To explore the Riemann Integral and Darboux Integral used in high school and university courses, proving their equivalence.
3. To compare the processes of deriving the Fundamental Theorem of Calculus (FTC) via Riemann and Henstock Integrals, illustrating the expansion of integrable functions.

Historical Development of the Integral

Early Foundations: Newton defined the integral as $\int_{a}^{b} f(x)dx = F(b) - F(a)$ for continuous functions. The history of integration stems from the ancient necessity to calculate areas of curved shapes.
Antiphon (c. 430 B.C.): The first to contribute to the area of a circle. He used the method of inscribing regular polygons and doubling the number of sides. As the number of sides increases, the area difference between the circle and the polygon approaches zero, providing an early glimpse of the limit concept.
Eudoxus (c. 408–355 B.C.): Developed the "Method of Exhaustion." He proved that for two circles with areas $A_1, A_2$ and diameters $d_1, d_2$ , the ratio is $A_1 : A_2 = d_1^2 : d_2^2$ . This method served as a tool for verification but was less effective for initial discovery.
Archimedes (c. 287–212 B.C.): The most advanced user of the exhaustion method. He calculated the area of a parabolic segment as $\frac{4}{3} \Delta ABC$ . He also discovered formulas for the surface area of a sphere ( $S = 4\pi r^2$ ) and its volume ( $V = \frac{2}{3} \cdot 2\pi r^3$ of the circumscribed cylinder). He used the "Method of Equilibrium" (a physical balancing approach) to find results before proving them rigorously.
Simon Stevin (1548–1620) and Luca Valerio (1552–1618): Attempted to avoid the cumbersome proofs by contradiction in the exhaustion method, utilizing rudimentary limit concepts for centers of gravity.
Johannes Kepler (1571–1630): Abandoned exhaustion for the concept that areas and volumes are the sum of infinitely many infinitely small parts. In Nova Stereometria Doliorum Vinariorum (1615), he calculated 84 types of volumes of revolution. He viewed a circle as a sum of infinite triangles with height equal to the radius $r$ , leading to Area $S = \frac{1}{2} r (2\pi r) = \pi r^2$ .
Bonaventura Cavalieri (1598–1647): Developed the "Geometry of Indivisibles" (1635). He considered areas the sum of lines and volumes the sum of planes. Cavalieri's Principle states that if two solids have equal cross-sectional areas at every height, they have equal volumes. He used this to calculate the area of an ellipse as $\pi ab$ .
Blaise Pascal (1623–1662): Replaced indivisibles with small rectangles to establish a more precise concept, influencing the development of differentials.
Pierre de Fermat (1601–1665): Anticipated differentiation through work on tangents and maxima/minima. He noted that near a maximum or minimum, function increments become infinitesimal.
John Wallis (1616–1703) and Isaac Barrow (1630–1677): Wallis systematized the results of Descartes and Cavalieri and introduced the infinity symbol $\infty$ . Barrow, Newton's teacher, was the first to realize that differentiation and integration are inverse operations, proving the Fundamental Theorem of Calculus in his Lectiones Geometricae.
Isaac Newton (1642–1727): Invented the Method of Fluxions. He treated variables as "fluents" ( $x, y, z$ ) changing with time and their velocities as "fluxions" ( $\dot{x}, \dot{y}, \dot{z}$ ). He applied calculus to motion, deriving distance from velocity and vice versa.
Gottfried Wilhelm von Leibniz (1646–1716): Independently developed calculus using indivisibles. He introduced the notations $dx, dy$ and the integral symbol $\int$ (an elongated S for summa). His notation proved more versatile and is still used today.
Augustin-Louis Cauchy (1789–1857): Provided the rigorous foundation for limits and continuity using the $\epsilon-\delta$ definition. He defined the integral for continuous functions.
Bernhard Riemann (1826–1866): Provided the first mathematically rigorous definition of the integral in 1854, expanding the range of integrable functions beyond continuity.
Henri Lebesgue (1875–1941): Introduced Measure Theory (1902) to define the "Lebesgue Integral," overcoming the limitations of the Riemann integral for more complex functions.
Ralph Henstock (1923–2007): Starting in the 1960s, developed a generalized Riemann integral (Henstock-Kurzweil integral) that simplifies the Fundamental Theorem of Calculus.

The Riemann Integral

Definitions

Partition (2.2.1): For an interval $I = [a, b]$ , a finite set $P = \{ x_0, x_1, \dots, x_n \}$ such that $a = x_0 < x_1 < \dots < x_n = b$ is a partition. The size (norm) of the partition is $\|P\| = \max\{x_i - x_{i-1}\}$ .
Riemann Sum (2.2.2): For a function $f: I \rightarrow \mathbb{R}$ , given a partition $P$ and selected points $\xi_i \in [x_{i-1}, x_i]$ , the Riemann sum is $S(P, f, \xi) = \sum_{i=1}^{n} f(\xi_i)(x_i - x_{i-1})$ .
Integrability (2.2.3): $f$ is Riemann integrable on $I$ if there exists a value $\gamma$ such that for any $\epsilon > 0$ , there exists $\delta > 0$ where $\|P\| < \delta \implies |S(P, f, \xi) - \gamma| < \epsilon$ . This is denoted as $\int_{a}^{b} f(t)dt$ .

Key Theorems

Cauchy Criterion (Theorem 2.2.1): $f$ is integrable iff for any $\epsilon > 0$ , partitions $P, Q$ with norm $< \delta$ satisfy $|S(P, f, \xi_1) - S(Q, f, \xi_2)| < \epsilon$ .
Continuity (Theorem 2.2.2): If $f$ is continuous on $[a, b]$ , it is Riemann integrable (using uniform continuity on compact sets).
Linearity and Additivity (Theorems 2.2.3 & 2.2.4):
1. $\int_{a}^{b} kf(t)dt = k \int_{a}^{b} f(t)dt$
2. $\int_{a}^{b} [f(t) + g(t)]dt = \int_{a}^{b} f(t)dt + \int_{a}^{b} g(t)dt$
3. $\int_{a}^{b} f(t)dt = \int_{a}^{c} f(t)dt + \int_{c}^{b} f(t)dt$ for $c \in (a, b)$ .
Fundamental Theorem of Calculus (FTC):
1. FTC 1 (Theorem 2.2.5): If $F$ is continuous, $F' = f$ , and $f$ is integrable, then $\int_{a}^{b} f(t)dt = F(b) - F(a)$ .
2. FTC 2 (Theorem 2.2.6): If $f$ is continuous and $F(x) = \int_{a}^{x} f(t)dt$ , then $F'(x) = f(x)$ .

Examples

Thomae-like Function: Let $f(x) = \frac{1}{p}$ if $x = \frac{q}{p}$ (coprime) and $0$ if $x$ is irrational. This function is Riemann integrable with an integral value of $0$ .
Dirichlet-like Function: Let $f(x) = x$ if $x$ is rational and $0$ if $x$ is irrational. This function is not Riemann integrable because the limit depends on whether $\xi_i$ are chosen as rational or irrational ( $\frac{1}{2}$ vs $0$ ).

The Darboux Integral

Definition and Refinement

Upper and Lower Sums (2.3.1): Defined using supremum $M(f, I_k)$ and infimum $m(f, I_k)$ over sub-intervals.
- Upper Sum: $U(f, P) = \sum M(f, I_k) \Delta x_k$
- Lower Sum: $L(f, P) = \sum m(f, I_k) \Delta x_k$
Integrability (2.3.2 & 2.3.3): A function is Darboux integrable if its Lower Integral $\sup(L^*)$ equals its Upper Integral $\inf(U^*)$ .
Darboux’s Criterion (Theorem 2.3.4): $f$ is integrable iff for any $\epsilon > 0$ , there exists $P$ such that $0 \leq U(f, P) - L(f, P) < \epsilon$ .

Equivalence to Riemann Integral

Theorem 2.3.5: Riemann integrability and Darboux integrability are equivalent. The values obtained are identical. The proof links Riemann sums $S(P, f, \xi)$ to the bounds $L(f, P) \leq S(P, f, \xi) \leq U(f, P)$ .

The Henstock Integral

Definitions and Theory

Tagged Partition (2.4.1): A partition where each sub-interval $[x_{i-1}, x_i]$ is associated with a point $\xi_i \in [x_{i-1}, x_i]$ .
Gauge (2.4.2): A positive function $\delta(x)$ on $[a, b]$ . A tagged partition is $\delta$ -fine if $[x_{i-1}, x_i] \subset (\xi_i - \delta(\xi_i), \xi_i + \delta(\xi_i))$ .
Cousin's Theorem (2.4.1): For any gauge $\delta(x)$ , there exists a $\delta$ -fine tagged partition. (Proven using the Nested Interval Theorem).
Henstock Integrability (2.4.3): $f$ is Henstock integrable if there exists $L$ such that for any $\epsilon > 0$ , there is a gauge $\delta(x)$ where all $\delta$ -fine partitions satisfy $|\sum f(\xi_i)(x_i - x_{i-1}) - L| < \epsilon$ .

Key Theorems and Properties

Measure Zero Sets (Theorem 2.4.2): If a function is $0$ almost everywhere (all points except a set of Lebesgue measure zero), its Henstock integral is $0$ .
General FTC (Theorem 2.4.3): If $F$ is continuous and differentiable everywhere except possibly on a countable set, then $F'$ is Henstock integrable and $\int_{a}^{b} F'(t)dt = F(b) - F(a)$ .
Saks-Henstock Lemma (Theorem 2.4.6): Provides detailed bounds on the sum of differences between function values and primitive values across sub-intervals, proving that if $f$ is Henstock integrable, its primitive $F(x)$ is continuous and $F' = f$ almost everywhere.
Comparison (Section 2.4): Every Riemann integrable function is Henstock integrable, but the reverse is not true. For example, the Dirichlet function ( $1$ on rationals, $0$ on irrationals) is Henstock integrable (value $0$ ) but not Riemann integrable.

Educational Implications and Conclusion

Historical Approach: Following the historical development (Genetic Principle) is effective for teaching. Starting from intuitive area concepts of the ancients and moving toward rigorous modern logic helps students bridge the gap between intuition and formal math.
Beyond Calculation: Teaching should avoid "rote calculation exams." Instead, lecturers should present integration as an expansion of human thought—from constants in Riemann ( $\|P\| < \delta$ ) to functional gauges in Henstock ( $\|P\| < \delta(x)$ ).
Mathematical Power: Exposure to the evolution of integration helps students experience the "flavor" of mathematics—searching for the unknown based on the known—and fosters the Mathematical Power intended by the curriculum.