Question

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

Answer 1

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

Answer 2

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

\(\Large ( 0, 0, M_{xy}/M) = (0,0, \frac 2 3 a ) \)

Answer 3

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