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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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    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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