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Is differential calculus a prerequisite for integral calculus? Because almost always you see that differential is taught before integral. Does that have a specific reason? Would it be recommended to start with integral calculus? Or is it an absolute non-go

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Even though is customary to learn differential calculus before integral calculus, some mathematicians think it's best to learn it the other way around. If you take a look at one of the best books on calculus, R. Courant's Differential and Integral Calculus, you'll find that the integral is defined before the derivative, and the construction of both branches is done somewhat simultaneously. –  Pragabhava Nov 26 '12 at 16:55
    
Can you tell me why some mathematicians think that? Or is it arbitrary? –  Jijbentlekker Nov 26 '12 at 16:57
    
Well, one reason might be that the notion of area and its summation is very intuitive, while differentiation is a whole new operation. –  Pragabhava Nov 26 '12 at 17:08

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The wonderful Analysis by Its History by Hairer and Wanner presents a first year calculus course by introducing the different topics in chronological order of their discovery. Of course this means that a fair bit of integration comes before some differentiation. The historical route is a great way to motivate the different definitions etc, and to see how mathematical knowledge is constructed/discovered. And its also a great way to learn calculus - it's not in the least pedagogically awful!

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Even though you could pretty well define, motivate and even write down integrals without using any differential calculus, you would lack means of effectively calculating interesting integrals without the Fundamental theorem. That's pedagogically awful!

On the other hand, derivatives are useful immediately (e.g. optimization problems) and given these, knowing anti-differentiation makes integration much more useful from the start.

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The proof of the fundamental theorem of calculus is avoided nine times out of ten. It is just technical... –  Peter Sheldrick Nov 26 '12 at 16:53
    
So with some very basic differential calculus you'd be set for starting with integral calculus? –  Jijbentlekker Nov 26 '12 at 16:56

I remember counting the squares of graph paper to estimate the area bounded by a curve well before I ever knew what the gradient of a tangent was. The ideas behind the integral can be taught to young children. However the problems of formalisation require some sophisticated knowledge of continuity and limit, which are normally encountered first, and in a simpler context, when dealing with the real numbers and derivatives of functions. Just think of all the different kinds of integral you know, with their different approaches to some of the critical issues of integration - and explain that to a youngster.

I wish I had been taught those original ideas by someone who really knew where they were leading - they could have planted motivations in my young mind for exploring ideas. There is perhaps a need for a brilliant book which links counting squares with the deeper ideas ...

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