What is Calculus?
Key Questions

If you're going into fields of science such as physics, chemistry, engineering, or higher mathematics, calculus is crucial. Calculus is the study of rates of change of things that algebra alone can't fully explain. Calculus is also linked very strongly to areas and volumes of shapes and solids.
In higher level mathematics, this concept translates to (say) finding areas and volumes of any solid, as well as quantifying various attributes of vector fields. Physicists use calculus (among other techniques) to work out the motion of moving things, and (perhaps most famously) the motion of planets and stellar bodies. Engineers use acceleration  a number not always easily obtained with dials  in their calculations of their designs, so that they can design objects, products, and structures that won't fall apart. And so on.
Calculus is mostly important in the sciences, but if you look around you, you can see other applications of calculus inside and outside your home.

Calculus is of outmost importance because of it's huge applicability.
Calculus is not restricted to mathematics and analysis, it's used pretty much everywhere  Physics, Chemistry, Economics, Biology, Engineering, Dynamic systems and so much more.
That's where the importance of calculus comes from.
Infinitesimal calculus as one might know, is the study of rate of change of functions with respect to variables the function is dependent on.
The notion of rate of change is so important.
What is the rate of change of prices ? What is the rate at which temperature of a body changes in heat conduction ? What is the rate of change of velocity of an object ? What is the rate of increase of the population under certain conditions ?You see, it's very important thus to study calculus. It has been the queen of mathematics and has been among the most important gifts the Mathematical world has given us.