Why We Never Actually Learn Riemann's Original Definition of Integrals - Riemann vs Darboux Integral
TLDRThe video script delves into the concept of integrals in mathematics, exploring their historical development from ancient ideas to the modern formalism by Newton and Leibniz. It discusses the Riemann and Darboux definitions of integrals, highlighting the limitations and complexities of Riemann's approach. The script simplifies the process of proving integrability using Darboux's method, demonstrating its practicality and ease of computation over Riemann's. The video also touches on the significance of supremum and infimum in refining the definition of integrals, ultimately showcasing the evolution and advancement in mathematical understanding of calculus.
Takeaways
- 📌 Integrals are mathematical tools used to calculate the area under a curve.
- 🌟 The concept of partitioning shapes into simpler figures like rectangles and triangles has ancient origins, but the modern integral was developed by Newton and Leibniz.
- 🔍 Riemann integrability refers to the condition where the limit of the area approximation using rectangles converges to the same value, regardless of how the rectangles are chosen.
- 📚 In introductory calculus, functions are assumed to be Riemann integrable, and the formal definition of Riemann integrability is often not covered.
- 🧩 The Darboux integral is an alternative to the Riemann integral that is theoretically equivalent but more practical in most cases.
- 🔎 Riemann's definition of integrals can be challenging to apply, especially for proving the integrability of simple functions.
- 📈 The process of proving Riemann integrability involves creating partitions and using tags to sample points within each subinterval.
- 🤔 Functions that are not Riemann integrable can be identified by showing that the limit of the Riemann sum does not exist, as demonstrated with a function that takes different values on rational and irrational inputs.
- 🛠️ The upper and lower sums of a partition provide bounds for the Riemann sum, and their limits as the partition becomes infinitesimally thin define the upper and lower integrals.
- 📊 The Darboux integral simplifies the process of proving integrability by focusing on the supremum (least upper bound) and infimum (greatest lower bound) of the upper and lower sums, respectively.
- 🎓 The video script emphasizes the importance of understanding foundational concepts in calculus, even if more advanced definitions and methods are used in practice.
Q & A
What is the integral in mathematics?
-The integral is a mathematical formalism invented to calculate the area under a curve. It approximates the area by partitioning the region into rectangles and triangles and summing up their areas.
Who are credited with the invention of the integral as we know it today?
-The integral as we know it today was invented by Sir Isaac Newton and Gottfried Wilhelm Leibniz, who independently developed the concept using rectangles to approximate the area under a curve.
What is Riemann integrability?
-Riemann integrability is the condition that the limit of the Riemann sum approaches the same value, no matter how the rectangles are chosen, for a given function on a certain interval. It is named after the mathematician Bernhard Riemann.
Why is the Riemann definition of integrals not popular in the education system?
-The Riemann definition of integrals is not popular in the education system because it can be cumbersome to prove the integrability of even simple functions using this definition, and there are more practical definitions like the Darboux integral that are easier to work with.
What is the Darboux integral and how is it related to the Riemann integral?
-The Darboux integral is a theoretical equivalent to the Riemann integral that defines the integral in terms of upper and lower sums, or the supremum and infimum of all possible Riemann sums for a given partition. It simplifies the process of proving integrability and is preferred in most mathematical contexts.
How does the Lebesgue integral differ from the Riemann and Darboux integrals?
-The Lebesgue integral is a fundamentally different approach to integration that measures the total length of a function at each height to calculate the area. It can handle functions that do not have discrete heights and is capable of digitizing the function into discrete heights for calculation.
What is a partition in the context of the Riemann integral?
-In the context of the Riemann integral, a partition is the process of dividing the interval from a to b into smaller subintervals. The collection of these subdivisions is called the partition P.
What are tags in the context of the generalized Riemann sum?
-Tags are the points sampled inside each subinterval of the partition to determine the height of each rectangle used in the Riemann sum. They can be chosen arbitrarily and are not limited to the endpoints or the midpoint of the subintervals.
How does the Riemann sum relate to the Riemann integral?
-The Riemann sum is the sum over all heights, which is the function evaluated at the sample point times the width of each rectangle. The Riemann integral is then defined as the limit of the Riemann sum as the norm of the partition (the length of the largest subinterval) approaches zero.
What is the significance of the telescoping sum in the proof of Riemann integrability?
-The telescoping sum is a technique used in the proof of Riemann integrability where a series of terms cancels out, leaving only the first and the last terms. This simplification helps in establishing the limit of the Riemann sum and proving the integrability of a function.
How does the concept of supremum and infimum help in the Darboux integral?
-The supremum and infimum are used in the Darboux integral to define the upper and lower sums, respectively. They ensure that the maximum and minimum values are accounted for, even in cases where the function has discontinuities or the maximum or minimum does not exist.
Outlines
📚 Introduction to Integrals and Riemann Integrability
This paragraph introduces the concept of integrals as a mathematical formalism for calculating the area under a curve. It discusses the historical development of integrals, highlighting the contributions of Newton, Leibniz, Riemann, Darboux, and Lebesgue. The paragraph emphasizes the Riemann integral, which uses rectangles to approximate the area and the concept of Riemann integrability, which is not often taught in elementary calculus classes. It also touches on the limitations of Riemann's definition and its unpopularity in education due to its complexity and the introduction of Darboux's definition, which is theoretically equivalent but more practical.
🔍 Examining Riemann Integrability and Non-Integrable Functions
This paragraph delves deeper into the Riemann integrability by discussing the generalized Riemann sum and its components, such as partitions, subintervals, and tags. It explains how the Riemann sum is calculated and how Riemann integrability is defined. The paragraph then provides an example of a non-Riemann integrable function, which takes different values for rational and irrational inputs. It also describes the technique used to prove that a limit does not exist, which is crucial for understanding when a function is not Riemann integrable.
🧠 Proving Riemann Integrability of a Simple Function
This paragraph focuses on the challenge of proving the Riemann integrability of a simple function, specifically x cubed, using Riemann's definition. It outlines the process of creating an arbitrary partition, choosing tags, and calculating the Riemann sum. The paragraph presents a detailed mathematical approach involving absolute values, substitutions, and the triangle inequality to show that the difference between the Riemann sum and the supposed limit approaches zero, thus proving the function's Riemann integrability.
🚀 Advantages of Darboux Integrability Over Riemann's Definition
This paragraph introduces Darboux's definition of integrability, which is theoretically equivalent to Riemann's definition but offers practical advantages. It explains the concepts of upper and lower sums, supremum, and infimum in the context of partitioning and tagging. The paragraph demonstrates how Darboux's definition simplifies the process of proving integrability, using the example of x cubed. It highlights the ease of computation and the applicability of high school-level mathematical techniques, making Darboux's definition the preferred method for establishing integrability in most cases.
Mindmap
Keywords
💡Integral
💡Newton and Leibniz
💡Riemann integrability
💡Darboux integrability
💡Partition
💡Tags
💡Telescoping sum
💡Supremum and Infimum
💡Monotone Convergence theorem
💡Equipartition
💡Theorems of integrability
Highlights
Integrals are mathematical formalisms used to calculate the area under a curve.
The concept of partitioning shapes into simpler figures like rectangles and triangles has existed since ancient times.
The integral as we know it today was co-invented by Newton and Leibniz, utilizing rectangles to approximate areas.
Riemann integrability refers to the condition where the limit of the area approximation approaches the same value regardless of the choice of rectangles.
In elementary calculus, all functions encountered are automatically Riemann integrable, eliminating the need for abstract details.
Darboux integrability is theoretically equivalent to Riemann's but is preferred due to its practical advantages.
Bolzano's definition of limit was the first rigorous definition in calculus, leading to Cauchy's formal definition of a derivative.
Riemann's definition of integrals was established about half a century after Cauchy's work.
Lebesgue introduced a completely new way to define the integral, measuring the total length of a function at each height.
The generalized Riemann sum allows for non-uniform partitioning of a region and the selection of arbitrary tags within each interval.
The Riemann sum is the sum of f evaluated at sample points, multiplied by the width of each rectangle delta.
A function is Riemann integrable on an interval if the limit of the Riemann sum approaches a particular value as the largest subinterval approaches zero.
The Riemann integral of a function is the limit of the Riemann sum, which represents the exact area under the curve.
The Riemann definition of integrability can be challenging to demonstrate for even simple functions, such as x cubed.
The upper and lower sums of a partition provide upper and lower bounds for the Riemann sum, simplifying the process of proving integrability.
Darboux's definition of integrability, involving upper and lower integrals, is a significant improvement over Riemann's, making calculations more manageable.
The difference between the upper and lower integral converges to zero, proving the function's integrability without prior knowledge of the limit.
The use of supremum and infimum in Darboux's definition ensures the existence of upper and lower sums even for functions with discontinuities.
The transition from Riemann to Darboux's definition illustrates the evolution of mathematical concepts for improved practical application.
Transcripts
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