Calculus is one of the important branches in Mathematics. Calculus is derived from the Latin words “small stone”. It simply means that by looking at the small pieces, we understand something on a larger scale. Differentiation and integration are the two important concepts explained under calculus. Both concepts are used in wide applications and have different specifications.
Calculus was originally introduced during the 17th century in Europe by Isaac Newton and Gottfried Wilhelm Leibniz. It is used in many fields such as physics, computer science, engineering, business, economics, medicine, etc., and in many other fields where mathematical calculations are practised.
What are Differentiation and Integration?
Differentiation represents the changes in small sections whereas integration expresses the addition of different small sections. In Maths, differential calculus is equal to the rate of change of one quantity with respect to another. Integration is the process of integrating smaller changes to get the final result.
Applications of Differentiation
To find the maximum and minimum values of particular functions such as cost, profit, loss, strength, etc., we use derivatives. Also, derivatives are used to understand the behaviour of moving objects.
Velocity in physics is one of the biggest examples of differential calculus. Velocity is equal to the rate of change displacement with respect to time in a particular direction. The smaller changes in velocity are calculated by the method differentiation.
Also, when we drive a car, the forces exerted while taking a turn is defined by tangents and normals, determined by using derivatives.
For curvilinear motion, where we need to find the velocity and acceleration of the objects, moving in a cervical path, is determined by differential calculus.
Applications of Integrations
If we have to find the area, volume, displacements on a larger scale, which occur by collecting the small data, we use integral calculus. We can use the integration process to calculate such quantities which cannot be calculated singularly.
To determine the actual length of power cable, in electrical engineering, we use integration, so that we can connect two substations, miles away from each other.
Relation Between Differentiation and Integration
Integral calculus and differential calculus both are counterparts of each other. If we differentiate a function, then to retain it we can directly integrate the derivative of the function with respect to the independent variable. Let us understand with some examples.
- To find the displacement of the object, we can directly integrate the velocity with respect to time to get the result
- To find the change in velocity, we need to integrate the acceleration function with respect to time
In Maths, integrals are subdivided into two parts:
- Definite Integrals
- Indefinite Integrals
Definite integrals have limits defined, under which we need to integrate the function. Whereas, there are no limits of indefinite integrals. Maths integration has many formulas, using which we can simply solve the problem based on integrals. Similarly, there are various formulas to find the derivative of the functions.
By this article, we have come across the importance of learning calculus (differential and integral). Also, we have seen many applications related to both the concepts on a larger scale.