## anonymous one year ago Use cylindrical coordinates. Find the mass and center of mass of the S solid bounded by the paraboloid z = 6x2 + 6y2 and the plane z = a (a > 0) if S has constant density K

1. anonymous

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2. anonymous

We want the intersection of z = 6x^2 + 6y^2 z = a a = 6x^2 + 6y^2 a/6 = x^2 + y^2 [√(a/6) ] ^2 = x^2 + y^2 the region E is 0 <= r <= sqrt( a/6) 0 <= theta <= 2π 6r^2 <= z <= a

3. anonymous

The mass of the solid is $\large M = \iiint_E K dV = \int_{0}^{2 \pi} \int_{0}^{\sqrt{a/6}}\int_{6r^2}^{a} Kr~ dz ~ dr ~ d\theta = \frac{Ka^2\pi}{12}$ The moment about the xy plane is $\large M_{xy} = \iiint_E zK dV = \int_{0}^{2 \pi} \int_{0}^{\sqrt{a/6}}\int_{6r^2}^{a} Kz~r~ dz ~ dr ~ d\theta = \frac{ Ka^3\pi}{18}$ Similarly the moments about the xz plane and the yz plane are $$\Large M_{xz} = 0 \\\Large M_{yz} = 0$$ The center of mass coordinates are $$\Large ( 0, 0, M_{xy}/M) = (0,0, \frac 2 3 a )$$

4. anonymous

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