Differentiation and Integration both are fundamental concepts of Calculus, which no doubt is among the outstanding achievements of human knowledge. The concepts and implications of Calculus have helped man in many disciplines from astronomy, economics, history, physics to even biology.
And the basic principles of calculus implemented in all these broad fields are differentiation and integration. In relation to the application of these fundamental concepts every student no matter from which discipline is, must be aware of the basis of integration and differentiation.
In this article we would discuss why integration and differentiation is important for students to learn.
Integration
The term integration has a simple definition, “the process of finding out the integrals”. While an integral is the term that allocates the numbers to functions while describing units that forms as a result of joining minute quantities. The integrals are classified into two subcategories
- Double Integral
2. Triple integral
The Definite integrals are simply used to find the area under the curve while the indefinite integrals are a function the derivative of which is a given function or also called the antiderivatives. Together both these integrals serve to solve several problems of calculus. You can find the double Integrals and triple Integrals values with the help of the online tools like double integral calculator and triple integral calculator.
However, the main difference between both the terms is, the definite integral in its existence, is a real number value, while on the other hand antiderivative has unlimited number of functions which vary just by a constant.
The best thing about integration process is that most of its formulas to could be directly derived from their corresponding derivative formulas. While a few are there including integration by parts, trigonometric substitution, change and substitution of variables and trigonometric variables whose formula needs to be derived by expending some extra calculations and derivations.
The integration helps us in finding the following real life quantities:
Area between curves
Distance, Velocity, Acceleration
Volume
The average value of a function
Work
Center of Mass
Kinetic energy; improper integrals
Probability
Arc Length
Surface Area
Differentiation
Finding out the function of rate of change of any variable in relation to other is known as the process of differentiation. Generally, integration process is used to find the rate of change for instance if we need to find the rate of change of velocity (acceleration) with respect time we require to implement the rules of differentiation.
If we explain it in terms of mathematics, that implies the process of finding the rate of change of x with respect to y is the differentiation. This process requires several rules to differentiate these functions.
While the derivative calculates the change in the output (function) value in relation to change in the input value. For instance, to calculate how rapidly an object changes its position with increase in the time we would say in terms of derivatives calculating its velocity.
When a derivative of a single-variable function occurs at a given input value, it is the slope of the tangent line to the function’s graph at that point. The tangent line is the function’s best linear approximation near the input value.
Also, a constant function would always have a zero as a derivative.
In real life differentiation help us in finding the rate of change of following quantities:
The rate of change in velocity with time, the acceleration
Finding the tangent and normal in a curve
Finding out the turning point
Calculating the highest and lowest point in a curve or graph
The main differentiation rules are: Sum and Difference Rule
Product Rule
Quotient Rule
Chain Rule