While it is obvious integration and differentiation aren’t related to each other, as one was caused by the tangent and one by the problems regarding area. However, it is found that the fundamental calculus theorem establishes a relationship between both the concepts.
In this article, we will explore the logical relationship between the two main concepts of calculus, the integration and the differentiation.
Fundamental Theorem of Calculus
The formula which defines the relationship between the integration and differentiation and gives a way for the assessment of a definite integral is called fundamental theorem of calculus (FTC). It is an important theorem as the logical relationship between differentiation and integration is explained and defined through it.
According to this theorem, if a function f has an antiderivative F, then the function’s definite integral from a to a to b is equivalent to F(b)-F (a). Furthermore, in discovering a function’s net change, average value or area the fundamental theorem of calculus is used. This theorem was found and investigated in the late 1600s and in the early 1700s by Sir Isaac Newton and Gottfried Wilhelm Leibniz respectively. Its name alone suggests how essential to the development of the calculus is this theorem.
The fundamental theorem of calculus comprises two parts, the second part of the fundamental theorem of calculus is known as the evaluation theorem. The first portion of the theorem refers to the pace at which an integral grows to integrate the function, which shows that integration and differentiation may be seen as reverse processes.
The second portion of the theorem allows us to calculate certain integrals easily. The second part of the fundamental theorem of calculus asserts that by evaluating the anti-derivative at the endpoints of the interval and subtraction, we are able to evaluate the integral if we are finding an antiderivative for the integrand. However, the link between differentiation and integration is based on the first part of this theorem.
Moreover in integrals, there are many types of integrals. One of the main types of integrals are Definite integration and Indefinite integration. Definite integration is defined as the difference between the integral function’s values for a lower value A and upper value B of the independent variable x.
While Indefinite Integration is defined as an antiderivative of a function. or you can simply say as integration its the antiderivative of an integrand. They might seem difficult but they are not. You can also try definite integral calculator and indefinite integral calculator with steps for solving definite integrand and indefinite integrand.
Relationship between Integration and Differentiation
Differentiation is the procedure by which a curve’s rate of change is calculated in the calculus. And as a matter of fact the integration is nothing more than the inverse of differentiation. It adds up all of the little regions below a curve and identifies the entire area.
As we already knew the two basic elements of calculus are the differentiation and integration. Integral calculus and differential calculus are simply the reverse disciplines of each other. Although differential calculus is essentially about how something is divided to track changes. Conversely, the integral calculus sums all the parts together.
Differentiation concerns the determination of a derivative which takes into consideration one of its variables the instantaneous rate of function change. It addresses the constantly varying numbers. Moreover, it is comparable to the slope of the tangent line, denoted by m=modification in y/modification at x.
Integration being the inverse of differentiation is hence often called anti-differentiation. The area between the bounds or limits under the function and the function could be determined if its derivative is given using the integration notion.
Because integration and differentiation are precisely the opposite, the integration can yield the original function, if derivatives have been identified. The basic theorem of calculus as discussed above describes this as well.
Conclusively, the logical relation between integration and differentiation is only that they are inverse of each other and related through the fundamental theorem of calculus. However, the differentiation all include differences and divisions, whereas integration all involves adding and averaging. To solve the problems of integration and derivation you can also use derivative calculators and integral calculators online to learn faster and better. As you will be able to understand all the processes of solving problems of integration and derivation.
Moreover, the differential determines the slope function when the distance between two points becomes less, the integration process also determines the area below the curve, as the number of rectangle divisions lying underneath the curve becomes big.