Using cylindrical coordinates compute $\int_B z \,dx \,dy \,dz$ where $B$ satisfies $1 \le z \le 2, x^2+y^2 \le 1$
My workings out:
I get the following for the integral $$\int_1^2z\,dz\int_0^{2\pi}d\theta\int_0^1r\,dr$$
I used the fact that $r^2 = x^2+y^2 \le 1 \implies r \le1$
Now this gives me the right answer, however I took a guess at the limits for $\theta$ so I was hoping for an explanation as to why its from $0 \le \theta \le 2\pi$ as I had thought it was from $0 \le \theta \le \frac{\pi}{2}$. Given that both values for $z$ and $r$ were positive.
Since $x=r\cos\theta$ and $y=r\sin\theta$, if you only take $\theta\in\left[0,\frac\pi2\right]$, then you will not get every pair $(x,y)$; you will only get those $(x,y)$ such that $x,y\geqslant0$. Since that cylinder has points all around the $z$-axis, you have to take $\theta\in[0,2\pi]$ or in any other closed interval whose length is $2\pi$.