원기둥·구면좌표 삼중적분

$E$ is located inside the cylinder ${x}^{2}+{y}^{2}=1$ and between the circular paraboloids $z=1-{x}^{2}-{y}^{2}$ and $z={x}^{2}+{y}^{2}.$

$E$ is located inside the sphere ${x}^{2}+{y}^{2}+{z}^{2}=1,$ above the $xy$-plane, and inside the circular cone $z=\sqrt{{x}^{2}+{y}^{2}}.$

$E$ is located inside the circular cone ${x}^{2}+{y}^{2}={(z-1)}^{2}$ and between the planes $z=0$ and $z=2.$

$E$ is located inside the cylinder ${x}^{2}+{y}^{2}=1$, above the circular cone $z=1-\sqrt{{x}^{2}+{y}^{2}}$, and below the circular paraboloid $z=1+{x}^{2}+{y}^{2}$, and between the planes $z=0$ and $z=2$.

[T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $\int_{-\pi /2}^{\pi /2}\,\int_{0}^{1}\,\int_{{r}^{2}}^{r}r\,dz\,dr\,d\theta .$ Find the volume $V$ of the solid. Round your answer to four decimal places.

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $\int_{0}^{\pi /2}\,\int_{0}^{1}\,\int_{{r}^{4}}^{r}r\,dz\,dr\,d\theta .$ Find the volume $V$ of the solid Round your answer to four decimal places.

Convert the integral $\int_{0}^{1}\,\int_{-\sqrt{1-{y}^{2}}}^{\sqrt{1-{y}^{2}}}\,\int_{{x}^{2}+{y}^{2}}^{\sqrt{{x}^{2}+{y}^{2}}}xz\,dz\,dx\,dy$ into an integral in cylindrical coordinates.

Convert the integral $\int_{0}^{2}\,\int_{0}^{y}\,\int_{0}^{1}(xy+z)dz\,dx\,dy$ into an integral in cylindrical coordinates.

In the following exercises, evaluate the triple integral $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=1,$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}+{z}^{2}\le 90,z\ge 0\}$

In the following exercises, evaluate the triple integral $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=1-\sqrt{{x}^{2}+{y}^{2}+{z}^{2}},$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}+{z}^{2}\le 9,y\ge 0,z\ge 0\}$

$f(x,y,z)=\sqrt{{x}^{2}+{y}^{2}},$ $B$ is bounded above by the half-sphere ${x}^{2}+{y}^{2}+{z}^{2}=9$ with $z\ge 0$ and below by the cone ${z}^{2}={x}^{2}+{y}^{2}.$

$f(x,y,z)=z,$ $B$ is bounded above by the half-sphere ${x}^{2}+{y}^{2}+{z}^{2}=16$ with $z\ge 0$ and below by the cone ${z}^{2}={x}^{2}+{y}^{2}.$

Show that if $F(\rho ,\theta ,\varphi )=f(\rho )g(\theta )h(\varphi )$ is a continuous function on the spherical box $B=\{(\rho ,\theta ,\varphi )|a\le \rho \le b,\alpha \le \theta \le \beta ,\gamma \le \varphi \le \psi \},$ then

$$\iiint_{B}F\,dV=\left(\int_{a}^{b}{\rho }^{2}f(\rho )dr\right)\left(\int_{\alpha }^{\beta }g(\theta )d\theta \right)\left(\int_{\gamma }^{\psi }h(\varphi )\sin \,\varphi \,d\varphi \right).$$

a. A function $F$ is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as $F(x,y,z)=f(\rho ),$ where $\rho =\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}.$ Show that $$\iiint_{B}F(x,y,z)dV=2\pi \int_{a}^{b}{\rho }^{2}f(\rho )d\rho ,$$

where $B$ is the region between the upper concentric hemispheres of radii $a$ and $b$ centered at the origin, with $0<a<b$ and $F$ a spherical function defined on $B.$ b. Use the previous result to show that $\iiint_{B}({x}^{2}+{y}^{2}+{z}^{2})\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\,dV=21\pi ,$ where $$B=\{(x,y,z)|1\le {x}^{2}+{y}^{2}+{z}^{2}\le 2,z\ge 0\}.$$

a. Let $B$ be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where $0<a<b.$ Consider F a function defined on B whose form in spherical coordinates $(\rho ,\theta ,\varphi )$ is $F(x,y,z)=f(\rho )\cos \,\varphi .$ Show that if $g(a)=g(b)=0$ and $\int_{a}^{b}h(\rho )d\rho =0,$ then $$\iiint_{B}F(x,y,z)dV=\frac{{\pi }^{2}}{4}[ah(a)-bh(b)],$$

where $g$ is an antiderivative of $f$ and $h$ is an antiderivative of $g.$ b. Use the previous result to show that $\iiint_{B}\frac{z\,\cos \sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}dV=\frac{3{\pi }^{2}}{2},$ where $B$ is the region between the upper concentric hemispheres of radii $\pi$ and $2\pi$ centered at the origin and situated in the first octant.

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in spherical coordinates. Let $f(\rho , \theta , \varphi )$ be the corresponding function in spherical coordinates b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to spherical coordinates and evaluate it.

$g(x,y,z)=z;$ $E=\{(x,y,z)|0\le {x}^{2}+{y}^{2}+{z}^{2}\le 1,z\ge 0\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in spherical coordinates. Let $f(\rho , \theta , \varphi )$ be the corresponding function in spherical coordinates b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to spherical coordinates and evaluate it.

$g(x,y,z)=x+y;$ $E=\{(x,y,z)|1\le {x}^{2}+{y}^{2}+{z}^{2}\le 4,z\ge 0,y\ge 0\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in spherical coordinates. Let $f(\rho , \theta , \varphi )$ be the corresponding function in spherical coordinates b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to spherical coordinates and evaluate it.

$g(x,y,z)=2xy;$ $E=\{(x,y,z)|\sqrt{{x}^{2}+{y}^{2}}\le z\le \sqrt{1-{x}^{2}-{y}^{2}},x\ge 0,y\ge 0\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in spherical coordinates. Let $f(\rho , \theta , \varphi )$ be the corresponding function in spherical coordinates b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to spherical coordinates and evaluate it.

$g(x,y,z)=z;$ $E=\{(x,y,z)|{x}^{2}+{y}^{2}+{z}^{2}-2z\le 0,\sqrt{{x}^{2}+{y}^{2}}\le z\}$

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E=\{(x,y,z)|\sqrt{{x}^{2}+{y}^{2}}\le z\le \sqrt{16-{x}^{2}-{y}^{2}},x\ge 0,y\ge 0\}$