Introducing integral in textbooks

  • #1
Hill
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I was very surprised to read the following in Needham, Visual Complex Analysis:

"It is therefore doubly puzzling that the Trapezoidal formula is taught in every introductory calculus course, while it appears that the midpoint Riemann sum RM is seldom even mentioned."

I was surprised because I clearly remember that in my school (long time ago, in a country far away) the midpoint approximation of a curve was the main visualization for integral while the trapezoidal one has been mentioned but deemed unnecessary.

I wonder if Needham is right and if so, why is it different?
 
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  • #2
We had Darboux, called it Riemann, and no trapezoid - and Lebesgue should have been in my opinion.
 
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  • #3
fresh_42 said:
We had Darboux, called it Riemann, and no trapezoid - and Lebesgue should have been in my opinion.
Why Lebesgue in an introductory calculus course of all places? Fwiw we did Darboux and called it Riemann too (trapezoid was part of a numerical methods course that mostly discussed better methods) but I don't think it helped me much...except in terms of gaining an appreciation for how these things are formalized I guess.
 
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haider said:
Why Lebesgue in an introductory calculus course of all places?
Well, firstly for the pun with yet another name. And I regret that I didn't add Archimedes to the row.

Secondly, I am of the opinion that we should start to teach mathematics at school, nowadays more than ever. WA can do all that silly algorithmic stuff we use to torture kids with. It has never been mathematics, it is algorithmics, maybe computing, or calculating, but definitely not mathematics. Darboux or Riemann or the trapezoid rule are all adjustments of Archimedes, 300 B.C. Just as if we hadn't developed mathematics in the meantime. Archimedes can be taught in 1 or 2 hours. It is the application of volumes. Lebesgue requires the understanding of volumes.

haider said:
Fwiw we did Darboux and called it Riemann too (trapezoid was part of a numerical methods course that mostly discussed better methods) but I don't think it helped me much...except in terms of gaining an appreciation for how these things are formalized I guess.
 
  • #5
Does anybody here possess Needham's book? I would like to see the entire quotation plus context plus chapter plus page.

"... the Trapezoidal formula is taught in every introductory calculus course ..." is right away wrong as two out of two answers show. And by the way: every and all are never true in real life. I really dislike such claims. So maybe we do Needham wrong here as we cannot verify the claim. And maybe Needham just used it as a rhetorical means for what he really wanted to say.
 
  • #6
fresh_42 said:
Does anybody here possess Needham's book? I would like to see the entire quotation plus context plus chapter plus page.

"... the Trapezoidal formula is taught in every introductory calculus course ..." is right away wrong as two out of two answers show. And by the way: every and all are never true in real life. I really dislike such claims. So maybe we do Needham wrong here as we cannot verify the claim. And maybe Needham just used it as a rhetorical means for what he really wanted to say.
IMG_0112.jpeg
IMG_0111.jpeg
 
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  • #7
So Needham is indeed to blame for that casual statement. I would have thought that a British would know that the world is a bit bigger than San Fransisco. Disappointing. Particularly disappointing is the fact that he does not distinguish between calculus and numerical mathematics. The methods that are used in a calculus book are completely irrelevant as both are simply ##O(x).##
 
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  • #8
fresh_42 said:
So Needham is indeed to blame for that casual statement.
Yes.
I enjoy reading his book, but he makes such statements from time to time.
How right is the following one?

1699733735524.png
 
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  • #9
What constitutes a "geometric" interpretation, one wonders?
 

1. What is the purpose of introducing integrals in textbooks?

The purpose of introducing integrals in textbooks is to teach students a fundamental concept in calculus that deals with accumulation and the area under curves. This concept is crucial for solving problems in various fields such as physics, engineering, economics, and beyond. Integrals help in understanding rates of change and total accumulation, which are essential concepts in both theoretical and applied mathematics.

2. At what educational level are integrals typically introduced in textbooks?

Integrals are typically introduced at the high school and college levels. In high school, they are usually part of an advanced mathematics or calculus course. At the college level, integrals are covered more comprehensively in calculus I and II courses, depending on the educational institution's curriculum.

3. How are integrals typically presented in textbooks?

In textbooks, integrals are usually introduced after the concept of derivatives. The presentation often starts with the fundamental theorem of calculus, which links the concept of differentiation and integration. Textbooks typically cover methods of integration, definite and indefinite integrals, and applications of integrals in real-world scenarios. Diagrams, step-by-step examples, and practice problems are commonly used to enhance understanding.

4. What are some common challenges students face when learning about integrals?

Some common challenges include understanding the abstract concept of accumulation, difficulty in visualizing the area under a curve, and mastering the various techniques of integration such as integration by parts, substitution, and partial fractions. Additionally, applying integrals to solve real-world problems can be challenging due to the complexity of setting up the appropriate integral to model the situation accurately.

5. How have approaches to teaching integrals evolved in textbooks over the years?

Over the years, approaches to teaching integrals in textbooks have evolved to include more interactive elements such as online tutorials, animated graphs, and software tools that allow students to visualize and manipulate curves and areas dynamically. There has also been an increased emphasis on applications in diverse fields to show the relevance of integrals. Furthermore, textbooks now often incorporate a more student-centered approach, with activities designed to engage learners in discovering the concepts of integration themselves through guided inquiry and problem-solving.

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