Integration as Reverse of Differentiation

Introduction

Integration is viewed as the reverse process of differentiation in calculus.
There are two main types of integration: indefinite integration and definite integration.
The result of integration is labelled as the integral.

Differentiation takes a function and provides its derivative, a function that tells the rate of change.
The integral of a derivative gives back the original function, showing the reversal aspect of integration.
For instance, if the derivative of a function F(x) is f(x), then the indefinite integral of f(x) with respect to x gives us F(x) plus a constant, C.

The usual notation for integration is the elongated ‘S’-like symbol ∫ followed by the function we want to integrate and the variable of integration such as dx. For example, ∫f(x) dx.
The added constant in indefinite integration is often referred to as the constant of integration.
The ‘a’ and ‘b’ at the top and bottom of the ∫ in definite integration are the limits of integration, indicating the interval over which we are integrating the function.

Knowledge of the basic integration formulae is essential, usually derived from the standard differentiation formulae.
The integral of a sum or difference of functions is equal to the sum or difference of their integrals respectively.
If a function is multiplied by a constant, the integral of the function is also multiplied by the same constant.
In order to integrate a function made by multiplying two others, techniques like the integration by parts or the use of identities are required.
The concept of substitution often simplifies integrations. It involves a change of variable to make the integration accessible.

Integration is applied in various scenarios including solving ordinary differential equations, calculating areas, volumes and also in physics to understand notions of work and energy.
Definite integration is used when calculating the area under a curve between two points.
It also allows us to compute average values of functions.
Use of definite integration techniques permits the calculation of arc lengths, surface area and volumes of revolution.

Forgetting to include the constant of integration in the indefinite integrations.
Neglecting to apply the basic rules correctly, for example, the power rule.
Troubles in the application of appropriate methodologies in cases of product or quotients of functions.

Serves as a critical tool in understanding and depicting continuous change.
Holds a significant place in nearly all areas of mathematics, natural sciences and engineering.
Understanding how the process of integration reverses differentiation is fundamental to calculus and numeric analysis.