# Multiple Integration using Cylindrical coordinates

The question is, find the volume of a region bounded by the paraboloid $z=1-4(x^2+y^2)$ and the $xy$-plane.

I have tried this question using the cylindrical coordinates and my answer comes out to be $\pi/8$ but that is incorrect the answer should come out to be $7\pi/12$.

I took the limits for $z$ from $0$ to $1-4r^2$, for $r$ from $0$ to $1/2$ and for $\theta$ from $0$ to $2\pi$.

Wonder where I went wrong. Help will be highly appreciated. I have checked the integration part that is correct, i think that the problem lies somewhere in taking the limits.

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How can we tell where you went wrong if you don't show us your work? – joriki Sep 22 '12 at 2:24
You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". Here's a basic tutorial and quick reference. There's an "edit" link under the question. – joriki Sep 22 '12 at 2:25
Did you use the Jacobian when changing the coordinates? – Sigur Sep 22 '12 at 2:37
Maple shows $\int_{0}^{2\pi}\int_{0}^{1/2}(r-4r^3)drd\theta=\pi/8$ – Sigur Sep 22 '12 at 2:41
There's no way it should be $7\pi/12$. Certainly the region is contained in the cylinder with $0 \le z \le 1$ and $r=\sqrt{x^2+y^2}\le 1/2$. That cylinder has height $1$ and radius $1/2$, so its volume is $\pi r^2 h = \pi/4 < 7\pi/12$. – mjqxxxx Sep 22 '12 at 3:16