Suppose we have polar coordinates $(r,\phi)$, where $r≥0$ and $0≤\phi<2\pi$.
I have a question about the upper boundary for $\phi$ when evaluating some double integrals: Why do we write $0≤\phi≤2\pi$ instead of $0≤\phi<2\pi$?
For example, if we are finding the area of the circle $x^2+y^2=1$ using a double integral, we say the bounds are $0≤r≤1$ and $0≤\phi≤2\pi$ to get $\int_0^{2\pi}\int_0^1r\,dr\,d\phi=\pi$. Why don't we write $0≤\phi<2\pi$?
Although it should not affect the result, I still want to know why we can include the case $\phi=2\pi$. Can we exclude this case?
Thank you.