2.7 Cylindrical and spherical coordinates  (Page 8/11)

$\left(3,0,\pi \right)$

$\left(1,\frac{\pi }{6},\frac{\pi }{6}\right)$

$\left(12,-\frac{\pi }{4},\frac{\pi }{4}\right)$

$\left(3,\frac{\pi }{4},\frac{\pi }{6}\right)$

$\left(4,0,0\right)$

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

$\left(0,3,0\right)$

$\left(\mathrm{-2},2\sqrt{3},4\right)$

[T] $\rho =3$

[T] $\phi =\frac{\pi }{3}$

[T] $\rho =2\text{cos}\phi$

[T] $\rho =4\text{csc}\phi$

[T] $\phi =\frac{\pi }{2}$

[T] $\rho =6\text{csc}\phi \text{sec}\theta$

$x^2+y^2-3z^2=0,$ $z\ne 0$

$x^2+y^2+z^2-4z=0$

$z=6$

$x^2+y^2=9$

[T] $\left(1,\frac{\pi }{4},3\right)$

[T] $\left(5,\pi ,12\right)$

$\left(3,\frac{\pi }{2},3\right)$

$\left(3,-\frac{\pi }{6},3\right)$

$\left(2,-\frac{\pi }{4},\frac{\pi }{2}\right)$

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

$\left(8,\frac{\pi }{3},\frac{\pi }{2}\right)$

$\left(9,-\frac{\pi }{6},\frac{\pi }{3}\right)$

The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length $a,$ where $a>0$

A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of $a$ and $b,$ respectively, where $b>a>0$

A solid inside sphere $x^2+y^2+z^2=9$ and outside cylinder ${\left(x-\frac{3}{2}\right)}^2+y^2=\frac{9}{4}$

A cylindrical shell of height $10$ determined by the region between two cylinders with the same center, parallel rulings, and radii of $2$ and $5,$ respectively

[T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid $z=x^2+y^2$ and cone $x^2+y^2-z^2=0.$

[T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy -plane between sphere $x^2+y^2+z^2=4$ and elliptical cone $x^2+y^2-z^2=0.$

Washington, DC, is located at $39\text{°}$ N and $77\text{°}$ W (see the following figure). Assume the radius of Earth is $4000$ mi. Express the location of Washington, DC, in spherical coordinates.

San Francisco is located at $37.78\text{°}\text{N}$ and $122.42\text{°}\text{W}.$ Assume the radius of Earth is $4000$ mi. Express the location of San Francisco in spherical coordinates.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $\left(4000,\text{−}43.17\text{°},102.91\text{°}\right).$

Find the latitude and longitude of Berlin if its spherical coordinates are $\left(4000,13.38\text{°},37.48\text{°}\right).$

[T] Consider the torus of equation ${\left(x^2+y^2+z^2+R^2-r^2\right)}^2=4R^2\left(x^2+y^2\right),$ where $R\ge r>0.$

1. Write the equation of the torus in spherical coordinates.
2. If $R=r,$ the surface is called a horn torus . Show that the equation of a horn torus in spherical coordinates is $\rho =2R\text{sin}\phi .$
3. Use a CAS to graph the horn torus with $R=r=2$ in spherical coordinates.