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I have a jordan domain $D = \{(x,y,z)| z + x^2 + y^2 \leq 1, z\geq 0\}$ and the Integral is $\int_D z$. My first step is to notice that $0 \leq z \leq 1 - x^2 - y^2$ so I'll integrate over $z$ first with $\int_0^{1-x^2 -y^2}z dz$ which becomes $\frac{1}{2}(1 - x^2 - y^2)^2$ and I want to integrate this over the circle of radius 1 (by definition of D) to get (after a jump to polar coord's) $\frac{1}{2}\int_0^1\int_0^{2\pi} (1-r^2)^2rd\theta dr$ which becomes $\pi \int_0^1 r-2 r^3+r^5 dr = \pi [\frac{r^2}{2}- \frac{r^4}{2}+ \frac{r^6}{6}]_0^{1}$ which is then $\frac{\pi}{6}$.

Does this look correct? and/or can anybody point out any mistakes?

Thanks in advance.

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    Looks fine. I cannot spot any mistake..2012-12-05

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The solution is correct. I have a minor comment: the integral $\pi \int_0^1 (1-r^2)^2r \,dr$ can be handled easier with $u=1-r^2$ substitution, which turns it into $\frac{\pi}{2} \int_0^1 u^2\,du = \frac{\pi}{6}$.