12.7E: Exercises for Cylindrical and Spherical Coordinates

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

For exercises 1 - 4, the cylindrical coordinates $(r,θ,z)$ of a point are given. Find the rectangular coordinates $(x,y,z)$ of the point.

1) $\left(4,\frac{π}{6},3\right)$

Answer

$(2\sqrt{3},2,3)$

2) $\left(3,\frac{π}{3},5\right)$

3) $\left(4,\frac{7π}{6},3\right)$

Answer

$(−2\sqrt{3},−2,3)$

4) $(2,π,−4)$

For exercises 5 - 8, the rectangular coordinates $(x,y,z)$ of a point are given. Find the cylindrical coordinates $(r,θ,z)$ of the point.

5) $(1,\sqrt{3},2)$

Answer

$\left(2,\frac{π}{3},2\right)$

6) $(1,1,5)$

7) $(3,−3,7)$

Answer

$\left(3\sqrt{2},−\frac{π}{4},7\right)$

8) $(−2\sqrt{2},2\sqrt{2},4)$

For exercises 9 - 16, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

9) [T] $r=4$

Answer

A cylinder of equation $x^2+y^2=16,$ with its center at the origin and rulings parallel to the $z$-axis,

10) [T] $z=r^2\cos^2θ$

11) [T] $r^2\cos(2θ)+z^2+1=0$

Answer

Hyperboloid of two sheets of equation $−x^2+y^2−z^2=1,$ with the $y$-axis as the axis of symmetry,

12) [T] $r=3\sin θ$

13) [T] $r=2\cos θ$

Answer

Cylinder of equation $x^2−2x+y^2=0,$ with a center at $(1,0,0)$ and radius $1$, with rulings parallel to the $z$-axis,

14) [T] $r^2+z^2=5$

15) [T] $r=2\sec θ$

Answer

Plane of equation $x=2,$

16) [T] $r=3\csc θ$

For exercises 17 - 22, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

17) $z=3$

Answer

$z=3$

18) $x=6$

19) $x^2+y^2+z^2=9$

Answer

$r^2+z^2=9$

20) $y=2x^2$

21) $x^2+y^2−16x=0$

Answer

$r=16\cos θ,\quad r=0$

22) $x^2+y^2−3\sqrt{x^2+y^2}+2=0$

For exercises 23 - 26, the spherical coordinates $(ρ,θ,φ)$ of a point are given. Find the rectangular coordinates $(x,y,z)$ of the point.

23) $(3,0,π)$

Answer

$(0,0,−3)$

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

25) $\left(12,−\frac{π}{4},\frac{π}{4}\right)$

Answer

$(6,−6,\sqrt{2})$

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

For exercises 27 - 30, the rectangular coordinates $(x,y,z)$ of a point are given. Find the spherical coordinates $(ρ,θ,φ)$ of the point. Express the measure of the angles in degrees rounded to the nearest integer.

27) $(4,0,0)$

Answer

$(4,0,90°)$

28) $(−1,2,1)$

29) $(0,3,0)$

Answer

$(3,90°,90°)$

30) $(−2,2\sqrt{3},4)$

For exercises 31 - 36, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

31) [T] $ρ=3$

Answer

Sphere of equation $x^2+y^2+z^2=9$ centered at the origin with radius $3$,

32) [T] $φ=\frac{π}{3}$

33) [T] $ρ=2\cos φ$

Answer

Sphere of equation $x^2+y^2+(z−1)^2=1$ centered at $(0,0,1)$ with radius $1$,

34) [T] $ρ=4\csc φ$

35) [T] $φ=\frac{π}{2}$

Answer

The $xy$-plane of equation $z=0,$

36) [T] $ρ=6\csc φ\sec θ$

For exercises 37 - 40, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.

37) $x^2+y^2−3z^2=0, \quad z≠0$

Answer

$φ=\frac{π}{3}$ or $φ=\frac{2π}{3};$ Elliptic cone

38) $x^2+y^2+z^2−4z=0$

39) $z=6$

Answer

$ρ\cos φ=6;$ Plane at $z=6$

40) $x^2+y^2=9$

For exercises 41 - 44, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φ in radians rounded to four decimal places.

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

Answer

$\left(\sqrt{10},\frac{π}{4},0.3218\right)$

42) [T] $(5,π,12)$

43) $\left(3,\frac{π}{2},3\right)$

Answer

$(3\sqrt{2},\frac{π}{2},\frac{π}{4})$

44) $\left(3,−\frac{π}{6},3\right)$

For exercises 45 - 48, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

45) $\left(2,−\frac{π}{4},\frac{π}{2}\right)$

Answer

$\left(2,−\frac{π}{4},0\right)$

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

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

Answer

$\left(8,\frac{π}{3},0\right)$

48) $\left(9,−\frac{π}{6},\frac{π}{3}\right)$

For exercises 49 - 52, find the most suitable system of coordinates to describe the solids.

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

Answer

Cartesian system, $\big\{(x,y,z)\,|\,0≤x≤a,\;0≤y≤a,\;0≤z≤a\big\}$

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

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

Answer

Cylindrical system, $\big\{(r,θ,z)\,|\,r^2+z^2≤9,\;r≥3\cos θ,\;0≤θ≤2π\big\}$

52) 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

53) [T] Use a CAS or CalcPlot3D to graph in cylindrical coordinates the region between elliptic paraboloid $z=x^2+y^2$ and cone $x^2+y^2−z^2=0.$

Answer

The region is described by the set of points $\big\{(r,θ,z)\,|\,0≤r≤1,\;0≤θ≤2π,\;r^2≤z≤r\big\}.$

54) [T] Use a CAS or CalcPlot3D 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 $x^2+y^2−z^2=0.$

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

Answer

$(4000,−77°,51°)$

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

57) Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $(4000,−43.17°,102.91°).$

Answer

$43.17°W, 22.91°S$

58) Find the latitude and longitude of Berlin if its spherical coordinates are $(4000,13.38°,37.48°).$

59) [T] Consider the torus of equation $\big(x^2+y^2+z^2+R^2−r^2\big)^2=4R^2(x^2+y^2),$ where $R≥r>0.$

a. Write the equation of the torus in spherical coordinates.

b. If $R=r,$ the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $ρ=2R\sin φ.$

c. Use a CAS or CalcPlot3D to graph the horn torus with $R=r=2$ in spherical coordinates.

Answer

a. $ρ=0, \quad ρ+R^2−r^2−2R\sin φ=0$

c.

60) [T] The “bumpy sphere” with an equation in spherical coordinates is $ρ=a+b\cos(mθ)\sin(nφ)$, with $θ∈[0,2π]$ and $φ∈[0,π]$, where $a$ and $b$ are positive numbers and $m$ and $n$ are positive integers, may be used in applied mathematics to model tumor growth.

a. Show that the “bumpy sphere” is contained inside a sphere of equation $ρ=a+b.$ Find the values of $θ$ and $φ$ at which the two surfaces intersect.

b. Use a CAS or CalcPlot3D to graph the surface for $a=14, b=2, m=4,$ and $n=6$ along with sphere $ρ=a+b.$

c. Find the equation of the intersection curve of the surface at b. with the cone $φ=\frac{π}{12}$. Graph the intersection curve in the plane of intersection.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

Links to CalcPlot3D added by Paul Seeburger (Monroe Community College).