Is differential calculus a prerequisite for integral calculus? 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
 A: 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.
A: 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!
A: 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 ...
A: In integration, we make use of differentiation, particularly when we are making substitutions.
While learning differentiation, we need not have the knowledge of differentiation.
Further, integration is called the reverse process of differentiation.
Hence, we have to learn differentiation before integration.
Differentiation helps one to understand a whole with micro-level approach to understand whole from inside out in the entire domain of a function, and it's mechanics is pathological analysis in nature. Integration is understanding of coverage of a given function in the Cartesian domain, in any co-ordinate definition system. Approach is reverse of approach to human understanding. Differential calculus is a child while integral calculus is grand parent. One first learns the evolution of child and then understands the old person. Differential coefficient, being tangent of inclination of function, is akin to psychology and behavioral pattern of the child; while integral calculus approach is the tendency of old generation to define it's domain boundaries.
It is, hence, essential to start understanding the child in all its ramifications before undertaking to understand of its higher generation in terms of latter's domain perceptions.
