I am trying to integrate over the region $E = E_{1} \cap E_{2} \cap E_{3}$, where $E_{1} = \{(x, y, z): y\geq0\}$, $E_{2} = \{(x, y, z): x^{2} + z^{2} \leq 4\}$, $E_{3} = \{(x, y, z): x+y+z \leq 3\}$.
Intuitively, this is a cylinder of radius 2 along the y-axis from $y=0$ until it is cut off by the plane defined by $E_{3}$. Which led me to my integral:
$\int_0^{2\pi}\int_0^2\int_0^{3-rsin(\theta)-rcos(\theta)} \!r\, \mathrm{d}y\mathrm{d}r\mathrm{d}\theta$
Which yields $12\pi$. My problem is, I get the same answer if I integrate:
$\int_0^{2\pi}\int_0^2\int_0^3 \!r\, \mathrm{d}y\mathrm{d}r\mathrm{d}\theta$
Clearly, the latter defines a larger space than the former, since the first one is the same cylinder but with a chunk sliced off. Thus, I have to be defining my integral wrong to be getting the same volume for both. Can someone show the the proper way to do this?