Calculus
Calculus is the branch of mathematics that is used to find the area of any geometric figure. It teaches how to measure the rate of change of functions and the area bounded by two or more functions.
Limits and induction are heavily used in calculus, to find integrals and derivatives. Most of calculus is based on the concepts of integrals and derivatives.
The use of calculus in pre-collegiate mathematics competitions is generally frowned upon. However, many physics competitions require it, as does the William Lowell Putnam competition.
The subject dealing with the rigorous foundations of calculus is called analysis, specifically real analysis.
