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7.5E:

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Exercise 7.5E.1

In the following exercises, evaluate the triple integrals Ef(x,y,z)dV over the solid E.

1.

f(x,y,z)=z,B={(x,y,z)|x2+y29,x0,y0,0z1}

A quarter section of a cylinder with height 1 and radius 3.

Answer
9π8

2. f(x,y,z)=xz2, B={(x,y,z)|x2+y216, x0, y0, 1z1}

3.

f(x,y,z)=xy, B={(x,y,z)|x2+y21, x0, xy, 1z1}

A wedge with radius 1, height 1, and angle pi/4.

Answer
18

4. f(x,y,z)=x2+y2, B={(x,y,z)|x2+y24, x0, xy, 0z3}

5. f(x,y,z)=ex2+y2, B={(x,y,z)|1x2+y24, y0, xy3, 2z3}

Answer

πe26

6. f(x,y,z)=x2+y2, B={(x,y,z)|1x2+y29, y0, 0z1}

Exercise 7.5E.2

7.

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 baˉf(r)dr=0 and ˉh(0)=0, where ˉf and ˉh are antiderivatives of f and h, respectively, show that

BF(x,y,z)dV=2πc(bˉf(b)aˉf(a))+π(b2a2)ˉh(c).

b. Use the previous result to show that

B(z+sinx2+y2)dx dy dz=6π2(π2),

where B is a cylindrical shell with inner radius π outer radius 2π, and height 2.

8.

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) f(z)\), where f, g, and h are differentiable functions. If ba˜f(r)dr=0, where ˜f is an antiderivative of f, show that

BF(x,y,z)dV=[b˜f(b)a˜f(a)][˜g(2π)˜g(0)][˜h(c)˜h(0)],

where ˜g and ˜h are antiderivatives of g and h, respectively.

b. Use the previous result to show that Bz sinx2+y2dx dy dz=12π2, where B is a cylindrical shell with inner radius π outer radius 2π, and height 2.

Exercise 7.5E.3

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates.

a. Express the region E in cylindrical coordinates.

b. Convert the integral Ef(x,y,z)dV to cylindrical coordinates.

9. E is bounded by the right circular cylinder r=4 sin θ, the rθ-plane, and the sphere r2+z2=16.

Answer

a. E={(r,θ,z)|0θπ, 0r4 sin θ, 0z16r2}

b. π04 sin θ016r20f(r,θ,z)r dz dr dθ

10. E is bounded by the right circular cylinder r=cos θ, the rθ-plane, and the sphere r2+z2=9.

11. E is located in the first octant and is bounded by the circular paraboloid z=93r2, the cylinder r=r, and the plane r(cos θ+sin θ)=20z.

Answer

a. \(E = \{(r,\theta,z) |0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq r \leq \sqrt{3}, \space 9 - r^2 \leq z \leq 10 - r(cos \space \theta + sin \space \theta)\};

b. π/203010r(cos θ+sin θ)9r2f(r,θ,z)r dz dr dθ

12. E is located in the first octant outside the circular paraboloid z=102r2 and inside the cylinder r=5 and is bounded also by the planes z=20 and θ=π4.

Exercise 7.5E.4

The following exercises give the function f and region E.

a. Express the region E and the function f in cylindrical coordinates.

b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it.

13. f(x,y,z)=1x+3, E={(x,y,z)|0x2+y29, x0, y0, 0zx+3}

Answer

a. E={(r,θ,z)|0r3, 0θπ2, 0zr cos θ+3},f(r,θ,z)=1r cos θ+3;

b. 30π/20r cos θ+30rr cos θ+3dz dθ dr=9π4

14. f(x,y,z)=x2+y2, E={(x,y,z)|0x2+y24, y0, 0z3x

15. f(x,y,z)=x, E={(x,y,z)|1y2+z29, 0x1y2z2}

Answer

a. y=r cos θ, z=r sin θ, x=z, E={(r,θ,z)|1r3, 0θ2π, 0z1r2}, f(r,θ,z)=z;

b. 312π01r20zr dz dθ dr=256π3

16. f(x,y,z)=y, E={(x,y,z)|1x2+z29, 0y1x2z2}

Exercise 7.5E.5

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

17. E is above the xy-plane, inside the cylinder x2+y2=1, and below the plane z=1.

Answer

π

18. E is below the plane z=1 and inside the paraboloid z=x2+y2.

19. E is bounded by the circular cone z=x2+y2 and z=1.

Answer

π3

20. E is located above the xy-plane, below z=1, outside the one-sheeted hyperboloid x2+y2z2=1, and inside the cylinder x2+y2=2.

21. E is located inside the cylinder x2+y2=1 and between the circular paraboloids z=1x2y2 and z=x2+y2.

Answer

π4

22. E is located inside the sphere x2+y2+z2=1, above the xy-plane, and inside the circular cone z=x2+y2.

23. E is located outside the circular cone x2+y2=(z1)2 and between the planes z=0 and z=2.

Answer

2π3

24. E is located outside the circular cone z=1x2+y2, above the xy-plane, below the circular paraboloid, and between the planes z=0 and z=2.

Exercise 7.5E.6

25. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates π/2π/210rr2rdzdrdθ. Find the volume V of the solid. Round your answer to four decimal places.

Answer

V=pi120.2618

A quarter section of an ellipsoid with width 2, height 1, and depth 1.

26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates π/2010rr4rdzdrdθ. Find the volume E of the solid. Round your answer to four decimal places.

Exercise 7.5E.7

27. Convert the integral 101y21z2x2+y2x2+y2xz dz dx dy into an integral in cylindrical coordinates.

Answer

\[\int_0^1 \int_0^{\pi} \int_{r^2}^r zr^2 \space cos \space \theta \space dz \space d\theta \space dr\

28. Convert the integral 20x010(xy+z)dz dx dy into an integral in cylindrical coordinates.

Exercise 7.5E.8

In the following exercises, evaluate the triple integral Bf(x,y,z)dV over the solid B.

29. f(x,y,z)=1, B={(x,y,z)|x2+y2+z290, z0}

30. f(x,y,z)=1, B={(x,y,z)|x2+y2+z290, z0}

A filled-in half-sphere with radius 3 times the square root of 10.

[Hide Solution]

Answer
180π10

31. f(x,y,z)=1x2+y2+z2, B={(x,y,z)|x2+y2+z29, y0, z0}

A quarter section of an ovoid with height 8, width 8 and length 18.

32. f(x,y,z)=x2+y2, B is bounded above by the half-sphere x2+y2+z2=9 with z0 and below by the cone z2=x2+y2.

Answer
81π(π2)16

33. f(x,y,z)=x2+y2, B is bounded above by the half-sphere x2+y2+z2=16 with z0 and below by the cone 2z2=x2+y2.

Exercise 7.5E.9

34. Show that if F(ρ,θ,φ)=f(ρ)g(θ)h(φ) is a continuous function on the spherical box B={(ρ,θ,φ)|aρb, αθβ, γφψ}, then BF dV=(baρ2f(ρ) dr)(βαg(θ) dθ)(ψγh(φ) sinφ dφ).

35. 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(ρ), where ρ=x2+y2+z2. Show that BF(x,y,z)dV=2πbaρ2f(ρ)dρ, 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.

Use the previous result to show that B(x2+y2+z2)x2+y2+z2dV=21π, where B={(x,y,z)|1x2+y2+z22, z0}.

36. 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 (ρ,θ,φ) is F(x,y,z)=f(ρ)cosφ. Show that if g(a)=g(b)=0 and bah(ρ)dρ=0, then BF(x,y,z)dV=π24[ah(a)bh(b)], where g is an antiderivative of f and h is an antiderivative of g.

Use the previous result to show that B=zcosx2+y2+z2x2+y2+z2dV=3π22, where B is the region between the upper concentric hemispheres of radii π and 2π centered at the origin and situated in the first octant.

Exercise 7.5E.10

The following exercises give the function f and region E.

a. Express the region E and function f in spherical coordinates.

b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it.

37. f(x,y,z)=z; E={(x,y,z)|0x2+y2+z21, z0}

38. f(x,y,z)=x+y; E={(x,y,z)|1x2+y2+z22, z0, y0}

Answer

a. f(ρ,θ,φ)=ρ sin φ (cos θ+sin θ), E={(ρ,θ,φ)|1ρ2, 0θπ, 0φπ2};

b. π0π/2021ρ3 sin2 φ(cos θ+sin θ) dρ dφ dθ=15π8

39. f(x,y,z)=2xy; E={(x,y,z)|x2+y2z1x2y2, x0, y0}

40. f(x,y,z)=z; E={(x,y,z)|x2+y2+z22z0, x2+y2z}

Answer

a. f(ρ,θ,φ)=ρ cos φ; E={(ρ,θ,φ)|0ρ2 cos φ, 0θ2π, 0φπ4};

b. 2π0π/402 cos φ0ρ3sin φ cos φ dρ dφ dθ=7π6

Exercise 7.5E.11

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

41. E={(x,y,z)|x2+y2z16x2y2, x0, y0}

42. E={(x,y,z)|x2+y2+z22z0, x2+y2z}

Answer

\(\pi\

 

)

43. Use spherical coordinates to find the volume of the solid situated outside the sphere ρ=1 and inside the sphere ρ=cos φ, with φ[0,π2].

44. Use spherical coordinates to find the volume of the ball ρ3 that is situated between the cones φ=π4 and φ=π3.

Answer

9π(21)

45. Convert the integral4416y216y216x2y216x2y2(x2+y2+z2)dzdxdy into an integral in spherical coordinates.

46. Convert the integral 4016x2016x2y216x2y2(x2+y2+z2)2dz dy dx into an integral in spherical coordinates.

Answer
π/20π/2040ρ6sinφdρdϕdθ

 

Exercise 7.5E.14

48. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates

ππ/2π/65π20ρ2sinφ dρ dφ dθ.

Find the volume V of the solid. Round your answer to three decimal places.

Answer

V=4π337.255

A sphere of radius 1 with a hole drilled into it of radius 0.5.

49. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as

2π0π/43π/410ρ2sinφ dρ dφ dθ.

Find the volume V of the solid. Round your answer to three decimal places.

50. [T] Use a CAS to evaluate the integral E(x2+y2)dV where E lies above the paraboloid

z=x2+y2 and below the plane z=3y.

Answer

343π32

51. [T]

a. Evaluate the integral Eex2+y2+z2dV, where E is bounded by spheres 4x2+4y2+4z2=1 and

x2+y2+z2=1.

b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

Exercise 7.5E.14

52. Express the volume of the solid inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=4as triple integrals

in cylindrical coordinates and spherical coordinates, respectively.

Answer
2π04216r216r2rdzdrdθ and
5π/6π/62π042cscϕρ2sinρdρdθdϕ
 

53. Express the volume of the solid inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=4 that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

 

 

 

 

 

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.

 


This page titled 7.5E: is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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