An integral calculus approach to the Basel problem

In this talk we present two proofs for the famous Basel problem which concerns Euler's formula for $\zeta(2)$. These two approaches are based on the papers of Stark (1978-https://www.jstor.org/stable/2320072) and Moreno (2016-https://www.tandfonline.com/doi/abs/10.4169/college.math.j.47.2.134). 

The main ingredients of these two proofs of $\zeta(2)=\frac{\pi^2}{6}$ are the first and the second mean value theorems for integrals which are taught in any introductory analysis course. 
If time allows, using similar ideas, we will also give a brief account for Euler's other two formulas such as $\zeta(4)=\frac{\pi^4}{90}$ and $\zeta(6)=\frac{\pi^6}{945}$.