7.5E:
( \newcommand{\kernel}{\mathrm{null}\,}\)
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+y2≤9,x≤0,y≤0,0≤z≤1}
- Answer
- 9π8
2. f(x,y,z)=xz2, B={(x,y,z)|x2+y2≤16, x≥0, y≤0, −1≤z≤1}
3.
f(x,y,z)=xy, B={(x,y,z)|x2+y2≤1, x≥0, x≥y, −1≤z≤1}
- Answer
- 18
4. f(x,y,z)=x2+y2, B={(x,y,z)|x2+y2≤4, x≥0, x≤y, 0≤z≤3}
5. f(x,y,z)=e√x2+y2, B={(x,y,z)|1≤x2+y2≤4, y≤0, x≤y√3, 2≤z≤3}
- Answer
-
πe26
6. f(x,y,z)=√x2+y2, B={(x,y,z)|1≤x2+y2≤9, y≤0, 0≤z≤1}
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(b2−a2)ˉh(c).
b. Use the previous result to show that
∭B(z+sin√x2+y2)dx dy dz=6π2(π−2),
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 sin√x2+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
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a. E={(r,θ,z)|0≤θ≤π, 0≤r≤4 sin θ, 0≤z≤√16−r2}
b. ∫π0∫4 sin θ0∫√16−r20f(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=9−3r2, the cylinder r=√r, and the plane r(cos θ+sin θ)=20−z.
- 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/20∫√30∫10−r(cos θ+sin θ)9−r2f(r,θ,z)r dz dr dθ
12. E is located in the first octant outside the circular paraboloid z=10−2r2 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)|0≤x2+y2≤9, x≥0, y≥0, 0≤z≤x+3}
- Answer
-
a. E={(r,θ,z)|0≤r≤3, 0≤θ≤π2, 0≤z≤r cos θ+3},f(r,θ,z)=1r cos θ+3;
b. ∫30∫π/20∫r cos θ+30rr cos θ+3dz dθ dr=9π4
14. f(x,y,z)=x2+y2, E={(x,y,z)|0≤x2+y2≤4, y≥0, 0≤z≤3−x
15. f(x,y,z)=x, E={(x,y,z)|1≤y2+z2≤9, 0≤x≤1−y2−z2}
- Answer
-
a. y=r cos θ, z=r sin θ, x=z, E={(r,θ,z)|1≤r≤3, 0≤θ≤2π, 0≤z≤1−r2}, f(r,θ,z)=z;
b. ∫31∫2π0∫1−r20zr dz dθ dr=256π3
16. f(x,y,z)=y, E={(x,y,z)|1≤x2+z2≤9, 0≤y≤1−x2−z2}
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+y2−z2=1, and inside the cylinder x2+y2=2.
21. E is located inside the cylinder x2+y2=1 and between the circular paraboloids z=1−x2−y2 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=(z−1)2 and between the planes z=0 and z=2.
- Answer
-
2π3
24. E is located outside the circular cone z=1−√x2+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/2−π/2∫10∫rr2rdzdrdθ. Find the volume V of the solid. Round your answer to four decimal places.
- Answer
-
V=pi12≈0.2618
26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates ∫π/20∫10∫rr4rdzdrdθ. Find the volume E of the solid. Round your answer to four decimal places.
Exercise 7.5E.7
27. Convert the integral ∫10∫√1−y2−√1−z2∫√x2+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 ∫20∫x0∫10(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+z2≤90, z≥0}
30. f(x,y,z)=1, B={(x,y,z)|x2+y2+z2≤90, z≥0}
[Hide Solution]
- Answer
- 180π√10
31. f(x,y,z)=1−√x2+y2+z2, B={(x,y,z)|x2+y2+z2≤9, y≥0, z≥0}
32. f(x,y,z)=√x2+y2, B is bounded above by the half-sphere x2+y2+z2=9 with z≥0 and below by the cone z2=x2+y2.
- Answer
- 81π(π−2)16
33. f(x,y,z)=√x2+y2, B is bounded above by the half-sphere x2+y2+z2=16 with z≥0 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)|a≤ρ≤b, α≤θ≤β, γ≤φ≤ψ}, 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=2π∫baρ2f(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)|1≤x2+y2+z2≤2, z≥0}.
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=zcos√x2+y2+z2√x2+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)|0≤x2+y2+z2≤1, z≥0}
38. f(x,y,z)=x+y; E={(x,y,z)|1≤x2+y2+z2≤2, z≥0, y≥0}
- Answer
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a. f(ρ,θ,φ)=ρ sin φ (cos θ+sin θ), E={(ρ,θ,φ)|1≤ρ≤2, 0≤θ≤π, 0≤φ≤π2};
b. ∫π0∫π/20∫21ρ3 sin2 φ(cos θ+sin θ) dρ dφ dθ=15π8
39. f(x,y,z)=2xy; E={(x,y,z)|√x2+y2≤z≤√1−x2−y2, x≥0, y≥0}
40. f(x,y,z)=z; E={(x,y,z)|x2+y2+z2−2z≤0, √x2+y2≤z}
- Answer
-
a. f(ρ,θ,φ)=ρ cos φ; E={(ρ,θ,φ)|0≤ρ≤2 cos φ, 0≤θ≤2π, 0≤φ≤π4};
b. ∫2π0∫π/40∫2 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+y2≤z≤√16−x2−y2, x≥0, y≥0}
42. E={(x,y,z)|x2+y2+z2−2z≤0, √x2+y2≤z}
- 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 ρ≤3 that is situated between the cones φ=π4 and φ=π3.
- Answer
-
9π(√2−1)
45. Convert the integral∫4−4∫√16−y2−√16−y2∫√16−x2−y2−√16−x2−y2(x2+y2+z2)dzdxdy into an integral in spherical coordinates.
46. Convert the integral ∫40∫√16−x20∫√16−x2−y2−√16−x2−y2(x2+y2+z2)2dz dy dx into an integral in spherical coordinates.
- Answer
- ∫π/20∫π/20∫40ρ6sinφdρdϕdθ
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/2∫π/65π∫20ρ2sinφ dρ dφ dθ.
Find the volume V of the solid. Round your answer to three decimal places.
- Answer
-
V=4π√33≈7.255
49. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as
∫2π0∫π/43π/4∫10ρ2sinφ 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 ∭Ee√x2+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
- ∫2π0∫42∫√16−r2−√16−r2rdzdrdθ and
- ∫5π/6π/6∫2π0∫42cscϕρ2sinρdρdθdϕ
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.