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August 11th, 2016, 05:57 PM   #11
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 Originally Posted by EvanJ Call the antiderivative at one point Q1 + C and the antiderivative at another point Q2 + C. To calculate the area under the curve in between the value of x that produced Q1 and the value of x that produced Q2, the area = (Q2 + C) - (Q1 + C).
One problem with this approach is that without defining definite integrals in terms of the area under a curve (usually via Riemann sums) and the Fundamental Theorem of Calculus, you have no justification for calling that quantity the area under the curve. This is a serious flaw when "the area under the curve" has such a strong natural meaning that we can verify answers against (given the right tools).

 August 24th, 2016, 06:37 AM #12 Senior Member   Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics In Hong Kong, Riemann summation is not usually taught until the first course in analysis. Teachers tend to start from indefinite integration, then teach the Fundamental Theorem of Calculus without proof.
 August 24th, 2016, 08:20 AM #13 Senior Member   Joined: Jun 2015 From: England Posts: 915 Thanks: 271 One thing forgotten by most teachers is that They are (I hope) familiar with the subject and its ramifications and development, whilst students are not. So looking at it from the point of view of a student, everything is new and unfamiliar. This doesn't apply only to calculus, it is pretty general. So when new material is presented the student glosses over much of the subtlety to get some much needed experience under his belt. When I first learned calculus both differentiation (first) and later integration were presented as limits. Then we quickly moved on in both cases to get familiarity with some examples / manipulation. This provided both comfort and motivation for pursuing the subject. Then still later the formal development of both procedures was revisited and connected. I don't think it really matters which comes first.
 August 24th, 2016, 08:26 AM #14 Newbie   Joined: Aug 2016 From: Viet Nam Posts: 1 Thanks: 0 I think 1 is better. Despite the fact that i was first introduced to integration as in 2
 August 24th, 2016, 12:06 PM #15 Newbie   Joined: Jul 2016 From: Switzerland Posts: 4 Thanks: 2 Math Focus: graph theory I was also introduced using method 2. I remember my teacher back then made us cut pieces of paper and try to fill an arbitrary shape with ever smaller pieces. Then we moved through the successive theorems up to the fundamental integral theorem. I think we saw only later the application of the integral as inverse of the derivative, or maybe in parrallel as exercises. I think if you go with method 2 first, you have to present things through an application in geometry or physics else the students won't relate.
 September 9th, 2016, 07:17 AM #16 Senior Member   Joined: Aug 2016 From: morocco Posts: 273 Thanks: 32 The best way to teach integration is to start by historic introduction : archimedes, Newton, Leibnitz, Cauchy and Riemann.

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