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 =_____.

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

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

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

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

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

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

$(2,-2,4)$

$( 2,0,4)$

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

$(0,3,4)$

$( 0,1,-3)$

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

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

$( 1,0,8)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\iiint\limits_{\kern-3ptE}xy{\it dV},$ 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 $x^{2}+y^{2}=1$.

$\iiint\limits_{\kern-3ptE}xy{\it dV},$ 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 $x^{2}+y^{2}=4$.

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

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

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

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

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

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

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.

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.

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.

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 $x^{2}+y^{2}=4$.

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

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 $z=x^{2}+y^{2}$.

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

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 $x^{2}+y^{2}+z^{2}=2$.

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 $x^{2}+z^{2}=1.$

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

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

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

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

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 $x^{2}+y^{2}=4z$.

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 $10\text{m}^{3}.$