Example 8.1.1

$R$ is the tetrahedron bounded by the planes $x\+3yplus;zequals;5$, $x\=3y$, $x\=0$, $z\=0$.

Example 8.1.2

$R$ is the tetrahedron cut from the first octant by the plane $3xplus;5yplus;7zequals;15$.

Example 8.1.3

$R$ is the region interior to the cylinder ${x}^{2}\+{y}^{2}\=9$ that is bounded below by the $\mathrm{xy}$plane, and above by the paraboloid $z\={x}^{2}\+{y}^{2}$.

Example 8.1.4

$R$ is the firstoctant region bounded by the planes $x\+z\=3$, and $y\+5zequals;15$.

Example 8.1.5

$R$ is the wedge the planes $z\=y$ and $z\=0$ cut from the cylinder ${x}^{2}\+{y}^{2}\=4$.

Example 8.1.6

$R$ is the firstoctant region bounded above by the cylinder $z\=3{x}^{2}$ and on the right by the paraboloid $3yequals;{x}^{2}plus;{z}^{2}$.

Example 8.1.7

$R$ is the region enclosed by the cylinder ${y}^{2}\+4{z}^{2}equals;16$ and the planes $x\=1$ and $x\+y\=5$.

Example 8.1.8

$R$ is the region common to the two cylinders ${x}^{2}\+{y}^{2}\=1$ and ${x}^{2}\+{z}^{2}\=1$.

Example 8.1.9

$R$ is the region enclosed by the cylinders $z\=4{x}^{2}$ and $z\=5{x}^{2}$, and the planes $y\=0$ and $x\+y\=\sqrt{2\/3}$.

Example 8.1.10

$R$ is the region that is inside the cylinder ${x}^{2}\+4{y}^{2}equals;4$, and that is bounded above and below by the planes $z\=x\+2$, and $z\=0$, respectively.

Example 8.1.11

$R$ is the region bounded by the elliptic paraboloids $z\=2{x}^{2}plus;5{y}^{2}$ and $z\=254{x}^{2}10{y}^{2}$.

Example 8.1.12

$R$ is the region in the upper halfplane that is above the cone ${z}^{2}\={x}^{2}\+{y}^{2}$ but below the sphere ${x}^{2}\+{y}^{2}\+{z}^{2}\=18$. (Use cylindrical coordinates.)

Example 8.1.13

$R$ is the region in the upper halfplane that is above the cone ${z}^{2}\={x}^{2}\+{y}^{2}$ but below the sphere ${x}^{2}\+{y}^{2}\+{z}^{2}\=18$. (Use spherical coordinates.)

Example 8.1.14

If $\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ are the variables in spherical coordinates, $R$ is the region bounded inside by the surface $\mathrm{\ρ}\=1\+\mathrm{cos}\left(\mathrm{\φ}\right)$ and outside by the sphere $\mathrm{\ρ}\=2$.

Example 8.1.15

$R$ is the region inside the cylinder ${x}^{2}\+{y}^{2}\=9$ and between the planes $z\=0$ and $y\+z\=5$.

Example 8.1.16

$R$ is the region that lies inside the sphere ${x}^{2}\+{y}^{2}\+{z}^{2}\=4$, and is between the cones $z\=\sqrt{{x}^{2}\+{y}^{2}}$ and $z\=\sqrt{3\left({x}^{2}\+{y}^{2}\right)}$.

Example 8.1.17

$R$ is the region that is bounded above by the paraboloid $z\=4{x}^{2}{y}^{2}$, below by the plane $z\=0$, and that lies outside the cylinder ${x}^{2}\+{y}^{2}\=1$.

Example 8.1.18

$R$ is the region bounded below by the paraboloid $z\={x}^{2}\+{y}^{2}$ and above by $z\=4x$.

Example 8.1.19

$R$ is the firstoctant region that lies between the cylinders $r\=1$ and $r\=3$, and that is bounded below by the plane $z\=0$ and above, by the surface $z\=1\+xy$.

Example 8.1.20

$R$ is the region that lies between the plane $z\=0$ and the paraboloid $z\=9{x}^{2}{y}^{2}$.

Example 8.1.21

$R$ is the region enclosed by the surface $\mathrm{\ρ}\=1\mathrm{cos}\left(\mathrm{\φ}\right)$, where $\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ are the variables in spherical coordinates.

Example 8.1.22

$R$ is the region that lies between the paraboloids $z\=4{x}^{2}{y}^{2}$ and $z\=3{x}^{2}plus;3{y}^{2}$.

Example 8.1.23

$R$ is the region that lies inside both the sphere ${x}^{2}\+{y}^{2}\+{z}^{2}\=16$ and the cylinder ${x}^{2}\+{y}^{2}4yequals;0$.

Example 8.1.24

$R$ is the region bounded above by the surface $z\=10y$, and below by the surface $z\=2{x}^{2}plus;3{y}^{2}$.

Example 8.1.25

$R$ is the region bounded above by the sphere ${x}^{2}\+{y}^{2}\+{z}^{2}\=6$, and below by the paraboloid $z\={x}^{2}\+{y}^{2}$.

Example 8.1.26

$R$ is the region interior to the surface $\mathrm{\ρ}\=2\mathrm{sin}\left(\mathrm{phi;}\right)$, where $\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ are the variables of spherical coordinates.

Example 8.1.27

$R$ is the firstoctant region enclosed by the cylinder ${x}^{2}\+{z}^{2}\=4$ and the plane $y\=3$.

Example 8.1.28

$R$ is the region bounded by the paraboloid $z\=4{x}^{2}{y}^{2}$ and the plane $z\=0$.

Example 8.1.29

$R$ is the firstoctant region that is bounded by the coordinate planes, and the additional planes $x\=1$, $x\+y\+z\=2$.

Example 8.1.30

$R$ is the region common to the three cylinders ${x}^{2}\+{y}^{2}\=1$, ${x}^{2}\+{z}^{2}\=1$, ${y}^{2}\+{z}^{2}\=1$.
