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# 2.7.7E: Exercises for Cylindrical and Spherical Coordinates

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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) $$\left(4,\frac{π}{6},3\right)$$

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

2) $$\left(3,\frac{π}{3},5\right)$$

3) $$\left(4,\frac{7π}{6},3\right)$$

$$(−2\sqrt{3},−2,3)$$

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

$$\left(2,\frac{π}{3},2\right)$$

6) $$(1,1,5)$$

7) $$(3,−3,7)$$

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

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] $$r=4$$

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] $$r^2\cos(2θ)+z^2+1=0$$

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] $$r=2\cos θ$$

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] $$r=2\sec θ$$

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) $$z=3$$

$$z=3$$

18) $$x=6$$

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

$$r^2+z^2=9$$

20) $$y=2x^2$$

21) $$x^2+y^2−16x=0$$

$$r=16\cos θ,\quad r=0$$

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) $$(3,0,π)$$

$$(0,0,−3)$$

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

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

$$(6,−6,\sqrt{2})$$

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) $$(4,0,0)$$

$$(4,0,90°)$$

28) $$(−1,2,1)$$

29) $$(0,3,0)$$

$$(3,90°,90°)$$

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] $$ρ=3$$

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] $$ρ=2\cos φ$$

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] $$φ=\frac{π}{2}$$

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) $$x^2+y^2−3z^2=0, \quad z≠0$$

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

38) $$x^2+y^2+z^2−4z=0$$

39) $$z=6$$

$$ρ\cos φ=6;$$ 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] $$\left(1,\frac{π}{4},3\right)$$

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

42) [T] $$(5,π,12)$$

43) $$\left(3,\frac{π}{2},3\right)$$

$$(3\sqrt{2},\frac{π}{2},\frac{π}{4})$$

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

$$\left(2,−\frac{π}{4},0\right)$$

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

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

$$\left(8,\frac{π}{3},0\right)$$

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 $$a>0$$

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 $$\left(x−\frac{3}{2}\right)^2+y^2=\frac{9}{4}$$

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 $$x^2+y^2−z^2=0.$$

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.

$$(4000,−77°,51°)$$

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 $$(4000,−43.17°,102.91°).$$

$$43.17°W, 22.91°S$$

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.

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.