Example 8.1.1

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

Example 8.1.2

$R$ is the tetrahedron cut from the first octant by the plane $3 x\+5 y\+7 z\=15$.

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 first-octant region bounded by the planes $x\+z\=3$, and $y\+5 z\=15$.

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 first-octant region bounded above by the cylinder $z\=3-x^2$ and on the right by the paraboloid $3 y\=x^2\+z^2$.

Example 8.1.7

$R$ is the region enclosed by the cylinder $y^2\+4 z^2\=16$ 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 $x^2\+4 y^2\=4$, 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 $z\=2 x^2\+5 y^2$ and $z\=25- 4 x^2-10 y^2$.

Example 8.1.12

$R$ is the region in the upper half-plane 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 half-plane 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 $\mathrm{\ρ}\=1\+\cos \left(\mathrm{\φ}\right)$ 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 $z\=4 x$.

Example 8.1.19

$R$ is the first-octant 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 $z\=1\+ x y$.

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 $\mathrm{\ρ}\=1-\cos \left(\mathrm{\φ}\right)$, 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 $z\=3 x^2\+3 y^2$.

Example 8.1.23

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

Example 8.1.24

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

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 $\mathrm{\ρ}\=2 \sin \left(\mathrm{\φ}\right)$, where $\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ are the variables of spherical coordinates.

Example 8.1.27

$R$ is the first-octant 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 first-octant 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 $x^2\+y^2\=1$, $x^2\+z^2\=1$, $y^2\+z^2\=1$.