http://en.wikipedia.org/wiki/History_of_calculus
Have a look at this wikipedia article. It has a great history of calculus.
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Just about all of calculus is based on differential and integral calculus, including Calculus 1! However, Calculus 1 is more likely to cover differential calculus, with integral calculus soon after. So there really isn't a right answer for this question.
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xx + sincos
Calc 2, then Calc 3, then usually Differential Equations
These are the general math courses in an undergraduate program of Mechanical Engineering. Actually, these are also the math courses required in ANY undergraduate Engineering curriculum: Algebra Trigonometry Analytic Geometry Differential Calculus Integral Calculus Mutivariable Calculus Differential Equations