Read the full paper at: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=49482 DOI: 10.4236/am.2014.516246 Author(s) Haifeng Xu, Jiuru Zhou Affiliation(s) School of Mathematical Sciences, Yangzhou University, Jiangsu, China. School of Mathematical Sciences, Yangzhou University, Jiangsu, China. ABSTRACT By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, we find that this special integral is two times of another special integral. By using this fact we give an easy way to calculate the value of the alternating sum of without using the Fourier expansion. Also, we discuss the relationship between Genocchi numbers and Bernoulli numbers and get some results about Bernoulli polynomials.eww140917gjr KEYWORDS Basel Problem, Zeta Function, Bernoulli Numbers, Bernoulli Polynomials