Calculus was start from the Greek Mathematicians. They started calculating the rise and run (slope) of curves and non straight lines.
Zeno of Elea in 455 BC, discovered the concept of Infinity.
Further contributions were made by Leucippus, Democritus, Antiphon and Eudoxus.
Archimedes, was the first important person who showed advance to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram.
Among other 'integrations' by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume of any segment of a paraboloid of revolution and a segment of an hyperboloid of revolution.
Other important contributors were Huygens, Leibniz, Newton, Jacob, Johann, and many more.
The father of Calculus in 19th Century is Cauchy, who completed the calculus and made the shape which today we see.
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http://en.wikipedia.org/wiki/History_of_calculus Have a look at this wikipedia article. It has a great history of calculus.
newton and Leibniz were first introduced the concept of limit independently
Calculus; by a long shot.
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.
It is certainly used in calculus, just as calculus can be used in trigonometry.