12.7E: Exercises for Cylindrical and Spherical Coordinates

Last updated
Save as PDF

Page ID: 20196

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

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

$CNX_Calc_Figure_12_07_201.jpg$

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,

$CNX_Calc_Figure_12_07_221.jpg$

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,

$CNX_Calc_Figure_12_07_223.jpg$

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,

$CNX_Calc_Figure_12_07_225.jpg$

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

15) [T] $ r=2\sec θ$

Answer

Plane of equation $ x=2,$

$CNX_Calc_Figure_12_07_227.jpg$

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$,

$CNX_Calc_Figure_12_07_229.jpg$

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$,

$CNX_Calc_Figure_12_07_031.jpg$

34) [T] $ ρ=4\csc φ$

35) [T] $ φ=\frac{π}{2}$

Answer

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

$CNX_Calc_Figure_12_07_233.jpg$

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\}.$

$CNX_Calc_Figure_12_07_235.jpg$

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.

$CNX_Calc_Figure_12_07_237.jpg$

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. $ρ=0, \quad ρ+R^2−r^2−2R\sin φ=0$

$CNX_Calc_Figure_12_07_238.jpg$

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).