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Integration Basic Concepts

INTRODUCTION:

The literal meaning of integration is to form something as a whole. Using the same analogy, the operation of integration in mathematics can be defined. A very common understanding is that integration of a given function f(x) is the total area below the curve f(x) that is the summation of various small area elements present below it.

Integration
  • Integration is also referred as anti-derivative since mathematically it is reverse of differentiation. As the rate of change of distance with respect to time (that is velocity) is calculated using differentiation, similarly, with integration one can calculate the distance analytically if the velocity and time relationship is given or graphically as the area under the velocity-time curve.



  • Integration is of two types:

    INDEFINITE INTEGRATION:
  • In this type of integration, the anti-derivative of the function is calculated directly, that is, without the limits of function using standard formulas and theorems of integration. The starting point and end point of function is not known.

  • In this integration, a very important aspect is of constant. Note that in differentiation the derivative of constant term becomes zero. So, while taking anti derivative, constant term is always added as the anti-derivative of zero.

  • DEFINITE INTEGRATION:
  • In this type of integration, the limits of integration are known. Hence to calculate the anti-derivative we can not only use standard formula and theorem but also properties of indefinite integration.

  • Also, note that the constant term which was absolutely necessary in indefinite integral vanishes here as it is added to both upper limit and lower limit and eventually it subtracted and becomes zero.



  • APPLICATIONS:

  • Like differentiation, Integration is also a very important mathematical tool. It has innumerous applications, few of which are mentioned below:

  • AREA UNDER CURVE:

    If the function of the curve is known then directly we can integrate and the answer would be nothing but area under the curve.



    VOLUME OF SOLID OF REVOLUTION:

    The fixed line about which the area is rotated is defined as axis of rotation and volume generated by rotating the area about the axis is known as the volume of solid revolution. If the function of the curve (whose integration would be the area under curve) is known then the volume of solid of revolution can be easily calculated as product of ∏ and integration of square of function .



    CENTROID OF AN AREA:

    The centroid is defined as the point at which whole mass of body seems to reside. To calculate it simply integrate the area elements which are multiplied with the distance from geometrical center and divide by total area. Integration here saves a lot of time as by definition we have summation of area elements with distance.



    AVERAGE VALUE OF FUNCTION:

    The average of function is summation of all the elements of function divided by the total number of element. So in a continuous function the summation can be calculated as the integration of the function within the defined limits.
    Further integration is used in advanced physics concepts, (eg: electromagnetics, theory of relativity, astronomy, acoustics, etc.) biology, economics, medicine, statistics and geology whose understanding is beyond the scope of this article.








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