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To convert a point \(P=\left( x,y,z\right) \) from rectangular coordinates to cylindrical coordinates \(\left( r,\theta ,z\right) \), use the equations: x =_____, y =_____, and z =_____.

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Expressed in cylindrical coordinates, the circular cone \( x^{2}+y^{2}=4z^{2}\) has the form _____.

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In cylindrical coordinates \(( r,\theta ,z) \), the differential \(dV\) of volume is _____

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True or False In cylindrical coordinates, \( \iiint\limits_{\kern-3ptE}\,r\,dr\,d\theta dz\) equals the volume of the solid \(E.\)

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\((-\sqrt{3},-1,-5)\)

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\((-1,\sqrt{3},4)\)

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\((1,1,\sqrt{2})\)

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\((2,-2,4)\)

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\(( 2,0,4) \)

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\(\left( -1,0,\dfrac{1}{2}\right) \)

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\((0,3,4) \)

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\(( 0,1,-3) \)

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\(\left( 2,\dfrac{\pi }{6},-5\right) \)

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\(\left( 4,\dfrac{\pi }{3},3\right) \)

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\(( 1,0,8) \)

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\(\left( 4,\dfrac{\pi }{6},2\right) \)

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\(\left( 2,\dfrac{\pi }{2},0\right) \)

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\( \left( -3,\dfrac{\pi }{2},1\right) \)

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\(\displaystyle\int_{-1}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}} }\int_{0}^{\sqrt{1-x^{2}-z^{2}}}dy\,dz\,dx\)

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\(\displaystyle\int_{0}^{1}\int_{0}^{\sqrt{1-z^{2}}}\int_{0}^{\sqrt{1-y^{2}-z^{2}} }{\it dx}\,{\it dy}\,{\it dz}\)

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\(\displaystyle\int_{0}^{2}\int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}}\int_{0}^{ \sqrt{x^{2}+y^{2}}}{\it dz}\,{\it dy}\,{\it dx}\)

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\(\displaystyle\int_{0}^{3}\int_{-\sqrt{9-x^{2}}}^{\sqrt{9-x^{2}}}\int_{0}^{\sqrt{ x^{2}+z^{2}}}{\it dy}\,{\it dz}\,{\it dx}\)

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\(\displaystyle\int_{\pi /6}^{\pi /2}\int_{0}^{3}\int_{0}^{r\sin \theta }r\csc ^{3}\theta \,{\it dz}\,dr\,d\theta \)

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\(\displaystyle\int_{\pi /6}^{\pi /2}\int_{0}^{1}\int_{0}^{\sin \theta }r\cos \theta \sin \theta \,{\it dz}\,dr\,d\theta \)

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\(\displaystyle\int_{0}^{\pi /3}\int_{0}^{1}\int_{0}^{e^{-1}}r\,{\it dz}\,dr\,d\theta \)

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\(\displaystyle\int_{0}^{\pi /3}\int_{0}^{\sin \theta }\int_{0}^{r\sin \theta }r\,{\it dz}\,dr\,d\theta \)

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\(\iiint\limits_{\kern-3ptE}{\it dV},\) where \(E\) is the solid enclosed by the planes \(z=1\) and \(z=4,\) and the cylinders \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=9\).

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\(\iiint\limits_{\kern-3ptE}{\it dV},\) where \(E\) is the solid enclosed by the \(xy\)-plane, \(z=3,\) and the cylinder \(x^{2}+y^{2}=4\).

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\(\iiint\limits_{\kern-3ptE}y{\it dV},\) where \(E\) is the solid enclosed by the planes \(z=1\) and \(z=x+3,\) and the cylinders \(x^{2}+y^{2}=1\) and \( x^{2}+y^{2}=4\).

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\(\iiint\limits_{\kern-3ptE}x{\it dV}\), where \(E\) is the solid enclosed by the planes \(z=0\) and \(z=x,\) and the cylinder \(x^{2}+y^{2}=9\).

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\(\iiint\limits_{\kern-3ptE}xy{\it dV},\) where \(E\) is the solid enclosed by the surfaces \(z=1-x-y\) and \(z=3-x-y\), whose projection onto the \(xy\)-plane is the circle \(x^{2}+y^{2}=1\).

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\(\iiint\limits_{\kern-3ptE}xy{\it dV},\) where \(E\) is the solid enclosed by the surfaces \(z=0\) and \(z=x^{2}+y^2\), whose projection onto the \(xy\)-plane is the circle \(x^{2}+y^{2}=4\).

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Volume Find the volume of the solid enclosed by the intersection of the sphere \(x^{2}+y^{2}+z^{2}=9\) and the cylinder \(x^{2}+y^{2}=2\).

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Volume Find the volume of the solid enclosed by the intersection of the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \( x^{2}+y^{2}=2x\).

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Volume Find the volume \(V\) of the solid enclosed by \( z=x^{2}+y^{2}\) and \(z=16-x^{2}-y^{2}\).

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Volume Find the volume \(V\) of the solid enclosed by \( z=x^{2}+y^{2}\) and \(z=2-x\).

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Volume Find the volume \(V\) of the solid enclosed by \(z^{2}=4x\) and \( x^{2}+y^{2}=2x.\)

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Mass Find the mass of a homogeneous solid of mass density \( \rho \) in the shape of a sphere of radius \(a\).

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Mass Find the mass of a solid in the shape of a sphere of radius \(a\), if the mass density \(\rho \) is proportional to the square of the distance from the center.

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Mass Find the mass \(M\) of an object in the shape of a right circular cylinder of height \(h\) and radius \(a\), if its mass density is proportional to the square of the distance from the axis of the cylinder.

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Moments of Inertia Find the moments of inertia \(I_{x}\) and \( I_{y}\) for the solid region enclosed by the hemisphere \(z=\sqrt{9-x^{2}-y^{2} }\) and the \(xy\)-plane, if the mass density is proportional to the distance from the \(xy\)-plane.

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Center of Mass Find the center of mass of a homogeneous solid in the first octant enclosed by the surface \(z=xy\) and the cylinder \( x^{2}+y^{2}=4\).

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Center of Mass Find the center of mass of a homogeneous solid enclosed by the surface \(x^{2}+y^{2}=4z\) and the plane \( z=2 \).

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Center of Mass Find the center of mass of a homogeneous solid enclosed by the inside of the sphere \(x^{2}+y^{2}+z^{2}=12\) and above the paraboloid \(z=x^{2}+y^{2}\).

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Center of Mass Find the center of mass of a homogeneous solid enclosed by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=4\).

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Mass Use cylindrical coordinates to find the mass of the homogeneous solid bounded on the sides by \(x^{2}+y^{2}=1\), on the bottom by the \(xy\)-plane, and on the top by \(x^{2}+y^{2}+z^{2}=2\).

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Joint Between Two Rods Find the mass of the intersection of two rods with constant mass density \(\rho \) that is formed by the cylinders \( x^{2}+y^{2}=1\) and \(x^{2}+z^{2}=1.\)

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Volume of a Mountain The height of a mountain (in \(\text{km}\)) can be approximated by \(z=5.3e^{-( x^{2}+y^{2}) }.\)

1. Sketch the mountain over the region \(x^{2}+y^{2}\leq 4.\)
2. Find the volume of the mountain over the region \(x^{2}+y^{2}\leq 4.\)

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\(\displaystyle\int_{0}^{2\pi }\int_{0}^{4}\int_{-r}^{\sqrt{16-r^{2}} }z^{2}r^{5}\cos ^{4}\theta \,{\it dz}\,dr\,d\theta \)

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\(\displaystyle\int_{0}^{\pi /2}\int_{0}^{2}\int_{-r^{2}}^{9}z^{2}r^{4}\sin ^{4}\theta \,{\it dz}\,dr\,d\theta \)

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Volume A circular hole of radius \(r\) is drilled through a sphere of radius \(R>r,\) as shown in the figure.

1. Find \(r\) in terms of \(R\) so that the hole removes exactly half of the volume of the sphere. (Hint: set up a volume integral in cylindrical coordinates for the hole.)
2. Find \(r\) rounded to three decimal places if \(R=10\text{cm}.\)

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Volume Find the volume enclosed on the top by the sphere \( x^{2}+y^{2}+z^{2}=5\) and on the bottom by the paraboloid \(x^{2}+y^{2}=4z\).

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Volume of a Joint Two pipes intersect at right angles as shown in the figure. Find the inner radius \(r\) of the pipes to ensure the volume of the intersecting joint is \(10\text{m}^{3}.\)