If your booster rocket is losing 100 kilograms of mass every second because
it's burning the fuel from its tanks, and the weight of the rocket's mass is
dereasing steadily because the force of gravity decreases steadily as you get
farther away from the center of the Earth, and you need to figure out how
much fuel you need, and which direction to aim the rocket, in order to have
exactly the right speed in the right direction when it reaches exactly the right
altitude to put a satellite into geostationary orbit so that everybody can aim
their little dishes at it and pick up 500 TV channels, THAT's where you need
Calculus. And I won't even go into how and why Calculus is used when people
want to fling a machine off the surface of the Earth and get it to the right
place to land on Mars 6 months later, or fly by Pluto 9 years later !
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Calculus is mainly about limits, which in turn are used to calculate the slope of a line (known as the "derivative"; lots of applications for that), and to calculate the area under a curve (the "integral" - also lots of applications for that). For more details, read the Wikipedia article on "Calculus", or read an introductory book on calculus. As prerequisites, you should be well-acquainted with high-school algebra.
Pretty much everything can be applied to calculus, and calculus to everything. If you have ever read the novel "Halo" you can see (in the first book) that Master Chief, had forgotten how much gravity was on the space-craft, and used calculus to determine the acceleration of gravity by timing the drop of a bolt and figuring out everything else. Calculus is especially important for any kind of profession that involves projectiles.
Vector calculus is applied in electrical engineering especially with the use of electromagnetics. It is also applied in fluid dynamics, as well as statics.
Integration can be used to calculate the area under a curve and the volume of solids of revolution.
well derivatives cannt be used without limits so it is application for calculus