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.