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**Hint:**In this question, we need to explain the importance of calculus in mathematics. At first we need to know the clear definition of calculus and the concepts of calculus. There are four main concepts in calculus . Mathematically calculus is the study of continuous change. Calculus is mainly devoted to the study of functions and limits. Calculus is originally called infinitesimal calculus or the calculus of infinitesimals . Mathematically , there are two types of calculus namely, single variable calculus and multi variable calculus. Here we need to explain the application of calculus in mathematics.

**Complete step by step answer:**

Here we need to explain why calculus is important.

Calculus is defined as the study of continuous change . The single variable calculus deals with only one function whereas multivariable calculus deals with the function with more than two variables. Mathematically calculus has two main branches , they are differential calculus and integral calculus.

Definition :

Differential calculus : Differential calculus is nothing but it cuts something into smaller pieces to see how it changes . Differential calculus is about differentiation . The term differentiation is nothing but it is a process of determining the derivative of a function at any point.

Integral calculus : Integral calculus is nothing but it joins smaller pieces together to see how much pieces are in. Integral calculus is about integration. The process of finding the integral is known as integration. The symbol ‘∫’ is the sign of integration. Mathematically, integrals are also used to find many useful quantities such as areas, volumes, displacement, etc.

These two branches are related together by the fundamental theorem of calculus.

Application of calculus :

Basically calculus is used in every science branch including physics , engineering, medicine, business, economy etc… There are four main concepts in calculus. There are limits, differentiation , integration and function theory. These concepts play a major role in solving the area of complicated shapes, in knowing the safety of vehicles, in evaluating survey data for business planning, credit card payment records, and in finding the changing conditions of a system that affect us etc… It helps to gain knowledge on a few concepts such as quadratic forms, in vector fields as derivative , in integration of forms, also in continuous differentiability , in tangent space and also in normal space and finally also in algebraic concepts. The application of multivariable calculus plays a major role in vector calculus which consists of topics like Green's theorem, Stroke theorem, Gradients, Divergence theorem, etc.

Thus calculus is important in all the fields of science including mathematics.

Calculus is important in all the fields of science including mathematics. Generally , calculus is very useful in a lot of fields including mathematics. Many science fields like engineering, physical science , computer science, business, economics, and lots of other fields use calculus.

**Note:**Moreover, calculus is a language of mathematicians, physicians, economists, biologists, architects, medical experts, and it is often used by them and also in all the fields of science. Some of the concepts of science are beautifully modelled and explained , such as birth and death rates, derivatives , integrals , heat and light, motion, electricity etc… with the help of calculus. The Quantitative data in finance also uses multivariable calculus to estimate future trends in the stock market and is also used in optimal control of a continuous-time dynamic system

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