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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?2012-09-22
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    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](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference)'s a basic tutorial and quick reference. There's an "edit" link under the question.2012-09-22
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    Did you use the Jacobian when changing the coordinates?2012-09-22
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    Maple shows $\int_{0}^{2\pi}\int_{0}^{1/2}(r-4r^3)drd\theta=\pi/8$2012-09-22
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    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 $\pi r^2 h = \pi/4 < 7\pi/12$.2012-09-22
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    Your answer is correct: I haven't even been able to find a way to "mess-up" the calculation to get a result of $ \ \frac{7 \pi}{12} \ $ . You can even just take "horizontal slices" through the paraboloid with $ \ r^2 \ = \ \frac{1}{4} (1 \ - \ z) \ $ and integrate over the _height_ of the solid: $$ V \ = \ \int_0^1 \ \pi \ [r(z)]^2 \ \ dz \ \ = \ \ \frac{\pi}{4} \ \int^1_0 \ 1 \ - \ z \ \ dz \ = \ \frac{\pi}{4} \ \cdot \ \frac{1}{2} \ \ . $$ Perhaps the provided answer is misprinted and was intended for a different problem...2014-07-11

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