### Author Topic: Error of Approximation in Case of Definite Integrals  (Read 3133 times)

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##### Error of Approximation in Case of Definite Integrals
« on: April 23, 2011, 10:45:07 am »
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Author : Rajesh Kumar Sinha, Satya Narayan Mahto, Dhananjay Sharan
International Journal of Scientific & Engineering Research, IJSER - Volume 2, Issue 4, April-2011
ISSN 2229-5518

Abstract— This paper proposes a method for computation of error of approximation involved in case of evaluation of integrals of single variable. The error associated with a quadrature rule provides information with a difference of approximation. In numerical integration, approximation of any function is done through polynomial of suitable degree but shows a difference in their integration integrated over some interval. Such difference is error of approximation. Sometime, it is difficult to evaluate the integral by analytical methods Numerical Integration or Numerical Quadrature can be an alternative approach to solve such problems. As in other numerical techniques, it often results in approximate solution. The Integration can be performed on a continuous function on set of data.

Index Terms— Quadrature rule, Simpsons rule, Chebyshev polynomials, approximation, interpolation, error.

INTRODUCTION
TO evaluate the definite integral of a function that has no explicit antiderivative of whose antiderivative is not easy to obtain; the basic method involved in approximating is numerical quadrature .

The methodolo-gy for computing the antiderivative at a given point, the polynomial   approximating the function   generally oscillates about the function. This means that if   over estimates the function   in one interval then it would underestimate it in the next interval [5]. As a result, while the area is overestimated in one interval, it may be underestimated in the next interval so that the overall effect of error in the two intervals will be equal to the sum of their moduli, instead the effect of the error in one interval will be neutralized to some extent by the error is the next interval. Therefore, the estimated error in an integration formula may be unrealistically too high. In view to above discussed facts, the paper would reveal types of approximation following the condition ‘best’ approximation for a given function, concentrating mainly on polynomial approximation. For approximation, there is considered a polynomial of first degree such as   a good approximation to a given function for the interval (a, b).

2    PROPOSED METHOD
2.1 Reflection on Approximation
This section cover types of approximation following the condition ‘best’ approximation for a given func-tion, concentrating mainly on polynomial approxima-tion. In this for approximation, there is considered a polynomial of first degree such as ; a good approximation to a given continuous function for the interval (0, 1).
Under the assumption of given concept two following statements may be considered as,
The Taylor polynomial at   (assuming  ex-ists)