My usage is that which is universally accepted in any university math department. As a freshman in a math/science/engineering major, you take (or finish after beginning in high school) three semesters of Calculus, always referred to with a capital "C" to mean specifically Newton's calculus. Here you learn to calculate integrals and derivatives so that you can use them as a tool. If you major in math, then as a junior or senior you will take analysis. Here is where you rigorously study integrals and derivatives. An introductory analysis course typically begins by defining Cauchy sequences, and then defining the real numbers as equivalence classes of Cauchy sequences of rationals. Building on this foundation, you then construct the basic theory of continuous functions, derivatives, and Riemann integrals. Finally, you use this theory to prove that the transformation rules of Newton's calculus are sound.