On Class of Area Vanishing Functions on the Unit Interval

Abstract

Among several classes of Riemann integrable real valued functions we are interested in finding class of functions whose (signed) integral values over a compact interval vanishes. Determining such class of functions was one of the important objectives of this paper. We begin our quest by introducing Bernoulli numbers then extending them to Bernoulli polynomials. We observe that Bernoulli polynomials are generalized version of the most famous and notorious Bernoulli numbers introduced by Jacob Bernoulli in 1713. In particular, we see that Bernoulli numbers are simply the constant terms of Bernoulli polynomials. Bernoulli numbers and Bernoulli polynomials play very big role in analyzing several aspects of mathematics and they occur unexpectedly in several counting problems. We prove some interesting properties of Bernoulli polynomials which generate another class of functions having the property of zero area in [0,1]. In this paper, we try to establish that such class of functions are precisely the Bernoulli polynomials and prove that the Riemann integral of five categories of Bernoulli polynomials over the compact interval [0,1] is zero. The geometric meaning of this fact for special cases is explained through several figures which will provide better insight and understanding. This paper will also provide an scope for generalizing in the analysis of Riemann integration of Bernoulli polynomials not restricted to just the interval [0,1] but for any compact interval in the real line.