The answers all generally make valid points but I think they miss the real reason engineers take a pretty standard 2 year math curriculum: efficiency in learning the rest of their coursework.  The people that devised the original curricula were not interested in creating a "liberal arts" foundation where calculus would exercise your mind etc.  They wanted to train engineers, plain and simple.

But to train engineers, you need to teach them subjects like mechanics, fluids, waves, etc.  To learn those different topics efficiently, you need calculus and linear algebra.  Sure you can replace a calculus argument by devising some very clever, elementary argument, but it's much better to give ONE argument via calculus that encompasses a variety of cases. Same thing goes for linear algebra.  For example, the concept of whether the nullspace of a linear system is trivial or not ties together quite nicely with the analogous concept in linear ODEs.  

One could argue all day about whether learning this way makes a better engineer or not, but one thing is clear to anyone that's taught: this is a very efficient way of training engineers.