there is correct the expresion [math]\int^{-\pi+\epsilon}_{\pi-\epsilon} d\theta[/math]....where [math]\theta[/math] is a angular coordinate between [math](-\pi,\pi)[/math]....¿what means this?...

 

i believe that this mean that the angular coordinate theta runs from [math]\pi-\epsilon[/math] to

[math]-\pi+\epsilon[/math] in the sense anti clock (figure)

I see why you might say that, but no it is not like that. You can only take Reimann integrals over continuous domains, but because your theta goes from (-pi, pi), you're not allowed to jump from pi to -pi!

 

If you could do the reimann sum as you were saying, then you would necessarily have:

 

[math]

\int^{-\pi+\epsilon}_{\pi-\epsilon} d\theta = -\int^{\pi-\epsilon}_{-\pi+\epsilon} d\theta

[/math]

 

However, all is not lost because the region you highlighted is just:

 

[math]

\int^{-\pi+\epsilon}_{-\pi} d\theta + \int^{\pi}_{\pi-\epsilon} d\theta

[/math]