원기둥·구면좌표 삼중적분

a. Let $B$ be a cylindrical shell with inner radius $a,$ outer radius $b,$ and height $c,$ where $0<a<b$ and $c>0.$ Assume that a function $F$ defined on $B$ can be expressed in cylindrical coordinates as $F(x,y,z)=f(r)+h(z),$ where $f$ and $h$ are differentiable functions. If $\int_{a}^{b}\tilde{f}(r)dr=0$ and $\tilde{h}(0)=0,$ where $\tilde{f}$ and $\tilde{h}$ are antiderivatives of $f$ and $h,$ respectively, show that $$\iiint_{B}F(x,y,z)dV=2\pi c\left(b\tilde{f}(b)-a\tilde{f}(a)\right)+\pi ({b}^{2}-{a}^{2})\tilde{h}(c).$$ b. Use the previous result to show that $\iiint_{B}\left(z+\sin \sqrt{{x}^{2}+{y}^{2}}\right)dx\,dy\,dz=6{\pi }^{2}(\pi -2),$ where $B$ is a cylindrical shell with inner radius $\pi ,$ outer radius $2\pi ,$ and height $2.$

a. Let $B$ be a cylindrical shell with inner radius $a,$ outer radius $b,$ and height $c,$ where $0<a<b$ and $c>0.$ Assume that a function $F$ defined on $B$ can be expressed in cylindrical coordinates as $F(x,y,z)=f(r)g(\theta )h(z),$ where $f,g,\,\text{and}\,h$ are differentiable functions. If $\int_{a}^{b}\tilde{f}(r)dr=0,$ where $\tilde{f}$ is an antiderivative of $f,$ show that $$\iiint_{B}F(x,y,z)dV=\left[b\tilde{f}(b)-a\tilde{f}(a)\right]\,\left[\tilde{g}(2\pi )-\tilde{g}(0)\right]\,\left[\tilde{h}(c)-\tilde{h}(0)\right],$$

where $\tilde{g}$ and $\tilde{h}$ are antiderivatives of $g$ and $h,$ respectively. b. Use the previous result to show that $\iiint_{B}z\,\sin \sqrt{{x}^{2}+{y}^{2}}dx\,dy\,dz=-12{\pi }^{2},$ where $B$ is a cylindrical shell with inner radius $\pi ,$ outer radius $2\pi ,$ and height $2.$

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.

Using spherical coordinates and integration, show that the volume of the sphere of radius $1$ centred at the origin is $4\pi/3$.

Find the limit or show that it does not exist

$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$