I am an engineering student at the University of Rhode Island. Most schools that offer engineering as a major will require you to go up to Level III Calculus, as well as classes involving equations with multiple unknowns. Mechanical engineering is a very rigorous and difficult major, but if you study hard, you will do absolutely fine.
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Robotics makes use of electrical, mechanical and software engineering. These fields, plus others, make use of most areas of mathematics (from algebra to artificial intelligence to complex analysis).
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
Mechanical engineering usually deals with forces and their effects on materials. Forces are vectors and so, to study their effects you need to use vector calculus.
locating objects; calculating the trajectory of a projectile; developing weapons uses engineering which uses lots of math
The usage of trig. in Mechanical Engineering is extensive. Trig. is used in almost every branch of the subject, especially where quantitative answers are needed to real physical problems such as the size of stresses in structural parts or the volume and frequency of a vibration.