Trigonometric Fourier series

Hi,

I have the following $\displaystyle 2\pi$-periodic function:

$\displaystyle \displaystyle f(t)=\left\{\begin{array}{lll}{\pi} ,\,\,0 < t < 1\\{}\\0 ,\,\, 1 < t< 2\pi-1\\{}\\{\pi} ,\,\, 2\pi-1 < t< 2\pi\end{array}\right$

$\displaystyle \displaystyle{c_{0}={\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)dt$ (1)$\displaystyle \displaystyle\implies$

$\displaystyle \displaystyle\implies\displaystyle{c_{0}={\frac{1} {2\pi}(\int_{0}^{1}{\pi}dt+{\int_{2\pi-1}^{2\pi}{\pi}dt)$

Is this integral to compute $\displaystyle c_{0}$ correct using *(1)*? If not what are the limits? Any help or guidance would be appriciate it.