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I'm trying to find the volume of the region underneath $x+y+z=0$, within the cylinder $x^2+y^2=1$ and above the $xy$ plane.

I decided to do the integral in cylindrical coordinates.

$$\int_{{3\pi}/4}^{{7\pi}/4} \int_0^{1} \int_0^{-rcos\theta-rsin\theta} r \,dz\,drd\theta$$

Is this the correct way to do the problem?

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    $\begingroup$ I checked it a few times, and think your answer is correct. $\endgroup$ – DeepSea Nov 26 '16 at 19:01
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Yes, that's one way. Another way is to slice the volume by planes $x + y = -\sqrt2t$, where $0 \leq t \leq 1$. Such a plane is distant $t$ from the origin and cuts off a chord of length $2\sqrt{1 - t^2}$ in the circle. Hence the volume is $\int_0^1\sqrt2t\cdot2\sqrt{1 - t^2}dt$, which gives the same answer as your integral, namely $2\sqrt2/3$.

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