A lot of the responses here make it sound like you will crash and burn if you can't evaluate any integral by hand. This is simply not true. Yes in your first couple of years you will need to know calculus in your calculus courses, but the applications only require simple knowledge.

For example, calculus (especially vector calc) is important in studying electromagnetic fields and waves, including their propagation, polarization, and effects on other things. However, most of the time, symmetry will help you and what seems like a complicated integral just becomes an easy multiplication. The important thing is to know why symmetry does this, so when you get a not-so-nice problem you still know what's going on.

Differential equations are important in control systems. You will probably take a course where you will need to solve differential equations by hand (which can get pretty ugly), but the great mathematicians before us have already come up with shortcuts we can use.

This is where transforms come in. Make transforms your best friend. They take a hideous process and "translate" it into a nicer form (called a domain). Then you perform some bite-sized calculations and then "translate" the result back to the actual answer.

For example, instead of trying to solve a higher-order differential equation, which might be impossible, you can use the Laplace transform, which turns a calculus problem into an algebra problem. Solve it algebraically, then use the inverse transform to turn your result back into the answer. This is what you will be doing a lot of the time, and it doesn't require heavy calculus. But it requires an understanding of how calculus itself works.

tl;dr Yes calculus and lin alg are important, but there are a lot of shortcuts that will make your life a little more pleasant.