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14.4E: Triple Integrals (Exercises 2)

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Terms and Concepts

1. The strategy for establishing bounds for triple integrals is "from ________ to ________, then from ________ to ________ and then from ________ to ________."

Answer:
We integrate from surface to surface, then from curve to curve and then from point to point.

2. Give an informal interpretation of what QdV means.

Answer:
QdV = Volume of the solid region Q

3. Give two uses of triple integration.

Answer:
To compute total mass or average density of a solid object, given a density function or to compute the average temperature in a solid region or object.

4. If an object has a constant density δ and a volume V, what is its mass?

Answer:
It's mass is δV.

Volume of Solid Regions

In Exercises 5-8, two surfaces f1(x,y) and f2(x,y) and a region R in the xy-plane are given. Set up and evaluate the triple integral that represents the volume between these surfaces over R.

5. f1(x,y)=8x2y2,f2(x,y)=2x+y;
R is the square with corners (1,1) and (1,1).

6. f1(x,y)=x2+y2,f2(x,y)=x2y2;
R is the square with corners (0,0) and (2,3).

7. f1(x,y)=sinxcosy,f2(x,y)=cosxsiny+2;
R is the triangle with corners (0,0),(π,0) and (π,π).

8. f1(x,y)=2x2+2y2+3,f2(x,y)=6x2y2;
R is the circle x2+y2=1.

In Exercises 9-16, a domain D is described by its bounding surfaces, along with a graph. Set up the triple integral that gives the volume of D in the indicated order of integration, and evaluate the triple integral to find this volume.

9. D is bounded by the coordinate planes and z=223x2y.
Evaluate the triple integral with order dzdydx.
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10. D is bounded by the planes y=0,y=2,x=1,z=0 and z=(2x)/2.
Evaluate the triple integral with order dxdydz.
13610.PNG

11. D is bounded by the planes x=0,x=2,z=y and by z=y2/2.
Evaluate the triple integral with order dydzdx.
13611.PNG

12. D is bounded by the planes z=0,y=9,x=0 and by z=y29x2.
Do not evaluate any triple integral. Just set this one up.
13612.PNG

13. D is bounded by the planes x=2,y=1,z=0 and z=2x+4y4.
Evaluate the triple integral with order dxdydz.
13613.PNG

14. D is bounded by the plane z=2y and by y=4x2.
Evaluate the triple integral with order dzdydx.
13614.PNG

15. D is bounded by the coordinate planes and y=1x2 and y=1z2.
Do not evaluate any triple integral. Which order would be easier to evaluate: dzdydx or dydzdx? Explain why.
13615.PNG

16. D is bounded by the coordinate planes and by z=1y/3 and z=1x.
Evaluate the triple integral with order dxdydz.
13616.PNG

In Exercises 17-20, evaluate the triple integral.

17. π/2π/2π0π0(cosxsinysinz)dzdydx

18. 10x0x+y0(x+y+z)dzdydx

19. π010z0(sin(yz))dxdydz

20. π2πx3xy2y2(cosxsinysinz)dzdydx

In the following exercises, evaluate the triple integrals over the rectangular solid box B.

B(2x+3y2+4z3) dV, where B={(x,y,z)|0x1, 0y2, 0z3}

[Hide Solution]

192

B(xy+yz+xz) dV, where B={(x,y,z)|1x2, 0y2, 1z3}

B(x cos y+z) dV, where B={(x,y,z)|0x1, 0yπ, 1z1}

[Hide solution]

0

B(z sin x+y2) dV, where B={(x,y,z)|0xπ, 0y1, 1z2}

In the following exercises, change the order of integration by integrating first with respect to z, then x, then y.

102132(x2+ln y+z) dx dy dz

[Hide Solution]

102132(x2+ln y+z) dx dy dz=356+2 ln2

101130(zex+2y) dx dy dz

213140(x2z+1y) dx dy dz

[Hide solution]

213140(x2z+1y) dx dy dz=64+12 ln 3

211210x+yz dx dy dz

Let F, G, and H be continuous functions on [a,b], [c,d], and [e,f], respectively, where a, b, c, d, e, and f are real numbers such that a<b, c<d, and e<f. Show that

badcfeF(x) G(y) H(z) dz dy dx=(baF(x) dx)(dcG(y) dy)(feH(z) dz).

Let F, G, and H be differential functions on [a,b], [c,d], and [e,f], respectively, where a, b, c, d, e, and f are real numbers such that a<b, c<d, and e<f. Show that

badcfeF(x) G(y) H(z) dz dy dx=[F(b)F(a)] [G(d)G(c)] H(f)H(e)].

In the following exercises, evaluate the triple integrals over the bounded region

E={(x,y,z)|axb, h1(x)yh2(x), ezf}.

E(2x+5y+7z) dV, where E={(x,y,z)|0x1, 0yx+1, 1z2}

[Hide solution]

7712

E(y ln x+z) dV, where E={(x,y,z)|1xe, 0yln x, 0z1}

E(sin x+sin y)dV, where E={(x,y,z)|0xπ2, cos xycos x, 1z1}

[Hide Solution]

2

E(xy+yz+xz)dV where E={(x,y,z)|0x1, x2yx2, 0z1}

In the following exercises, evaluate the triple integrals over the indicated bounded region E.

E(x+2yz) dV, where E={(x,y,z)|0x1, 0yx, 0z5xy}

[Hide Solution]

430120

E(x3+y3+z3) dV, where E={(x,y,z)|0x2, 0y2x, 0z4xy}

Ey dV, where E={(x,y,z)|1x1, 1x2y1x2, 0z1x2y2}

[Hide Solution]

0

Ex dV, where E={(x,y,z)|2x2, 41x2y4x2, 0z4x2y2}

In the following exercises, evaluate the triple integrals over the bounded region E of the form

E={(x,y,z)|g1(y)xg2(y), cyd, ezf}.

Ex2 dV, where E={(x,y,z)|1y2xy21, 1y1, 1z2}

[Hide Solution]

64105

E(sin x+y) dV, where E={(x,y,z)|y4xy4, 0y2, 0z4}

E(xyz) dV, where E={(x,y,z)|y6xy, 0y1x, 1z1}

[Hide Solution]

1126

Ez dV, where E={(x,y,z)|22yx2+y, 0y1x, 2z3}

In the following exercises, evaluate the triple integrals over the bounded region

E={(x,y,z)|g1(y)xg2(y), cyd, u1(x,y)zu2(x,y)}

Ez dV, where E={(x,y,z)|yxy, 0y1, 0z1x4y4}

[Hide Solution]

113450

E(xz+1) dV, where E={(x,y,z)|0xy, 0y2, 0z1x2y2}

E(xz) dV, where E={(x,y,z)|1y2xy, 0y12x, 0z1x2y2}

[Hide Solution]

1160(6341)

E(x+y) dV, where E={(x,y,z)|0x1y2, 0y1x, 0z1x}

In the following exercises, evaluate the triple integrals over the bounded region

E={(x,y,z)|(x,y)D, u1(x,y)xzu2(x,y)}, where D is the projection of E onto the xy-plane

D(21(x+y) dz) dA, where D={(x,y)|x2+y21}

[Hide Solution]

3π2

D(31x(z+1) dz) dA, where D={(x,y)|x2y21, x5}

D(10xy0(x+2z) dz) dA, where D={(x,y)|y0, x0, x+y10}

[Hide Solution]

1250

D(4x2+4y20y dz) dA, where D={(x,y)|x2+y24, y1, x0}

The solid E bounded by y2+z2=9, z=0, and x=5 is shown in the following figure. Evaluate the integral Ez dV by integrating first with respect to z, then y, and then x.

A solid arching shape that reaches its maximum along the y axis with z = 3. The shape reach zero at y = plus or minus 3, and the graph is truncated at x = 0 and 5.

[Hide Solution]

50339y20z dz dy dx=90

The solid E bounded by y=x, x=4, y=0, and z=1 is given in the following figure. Evaluate the integral Exyz dV by integrating first with respect to x, then y, and then z.

A quarter section of an oval cylinder with z from negative 2 to positive 1. The solid is bounded by y = 0 and x = 4, and the top of the shape runs from (0, 0, 1) to (4, 2, 1) in a gentle arc.

[T] The volume of a solid E is given by the integral 020xx2+y20dz dy dx. Use a computer algebra system (CAS) to graph E and find its volume. Round your answer to two decimal places.

[Hide Solution]

V=5.33

A complex shape that starts at the origin and reaches its maximum at (negative 2, negative 2, 8). The shape is truncated by the x = y plane, the x = 0 plane, the y = negative 2 plane, the z = 0 plane, and a complex triangular-like shape with curved edges and sides (negative 2, negative 2, 8), (0, 0, 0), and (0, negative 2, 4).

[T] The volume of a solid E is given by the integral 010x31+x2+y20dz dy dx. Use a CAS to graph E and find its volume V. Round your answer to two decimal places.

In the following exercises, use two circular permutations of the variables x, y, and z to write new integrals whose values equal the value of the original integral. A circular permutation of x, y, and z is the arrangement of the numbers in one of the following orders: y, z, and x or z, x, and y.

103142(x2z2+1)dx dy dz

[Hide Solution]

103142(y2z2+1)dz dx dy; 103142(x2z2+1)dx dy dz

3010x+10(2x+5y+7z)dy dx dz

10yy1x4y40ln xdz dx dy

1110yy6(x+yz)dx dy dz

Set up the integral that gives the volume of the solid E bounded by y2=x2+z2 and y=a2, where a>0.

[Hide Solution]

V=aaa2z2a2z2a2x2+z2dy dx dz

Set up the integral that gives the volume of the solid E bounded by x=y2+z2 and x=a2, where a>0.

Find the average value of the function f(x,y,z)=x+y+z over the parallelepiped determined by x+0, x=1, y=0, y=3, z=0, and z=5.

[Hide Solution]

92

Find the average value of the function f(x,y,z)=xyz over the solid E=[0,1]×[0,1]×[0,1] situated in the first octant.

Find the volume of the solid E that lies under the plane x+y+z=9 and whose projection onto the xy-plane is bounded by x=sin1y, y=0, and x=π2.

Consider the pyramid with the base in the xy-plane of [2,2]×[2,2] and the vertex at the point (0,0,8).

a. Show that the equations of the planes of the lateral faces of the pyramid are 4y+z=8, 4yz=8, 4x+z=8, and 4x+z=8.

b. Find the volume of the pyramid.

[Hide Solution]

a. Answers may vary; b. 1283

Consider the pyramid with the base in the xy-plane of [3,3]×[3,3] and the vertex at the point (0,0,9).

a. Show that the equations of the planes of the side faces of the pyramid are 3y+z=9, 3y+z=9, y=0 and x=0.

b. Find the volume of the pyramid.

The solid E bounded by the sphere of equation x2+y2+z2=r2 with r>0 and located in the first octant is represented in the following figure.

The eighth of a sphere of radius 2 with center at the origin for positive x, y, and z.

a. Write the triple integral that gives the volume of E by integrating first with respect to z, then with y, and then with x.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

[Hide Solution]

a. 40r2x20r2x2y20dz dy dx; b. 20r2x20r2x2y20dz dx dy,

r0r2x20r2x2y20dy dx dz, r0r2x20r2x2y20dy dz dx,

r0r2x20r2x2y20dx dy dz, r0r2x20r2x2y20dx dz dy,

The solid E bounded by the sphere of equation 9x2+4y2+z2=1 and located in the first octant is represented in the following figure.

In the first octant, a complex shape is shown that is roughly a solid ovoid with center the origin, height 1, width 0.5, and length 0.35.

a. Write the triple integral that gives the volume of E by integrating first with respect to z then with y and then with x.

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

Find the volume of the prism with vertices (0,0,0), (2,0,0), (2,3,0), (0,3,0), (0,0,1), and (2,0,1).

[Hide Solution]

3

Find the volume of the prism with vertices (0,0,0), (4,0,0), (4,6,0), (0,6,0), (0,0,1), and (4,0,1).

The solid E bounded by z=102xy and situated in the first octant is given in the following figure. Find the volume of the solid.

A tetrahedron bounded by the x y, y z, and x z planes and a triangle with vertices (0, 0, 10), (5, 0, 0), and (0, 10, 0).

[Hide Solution]

2503

The solid E bounded by z=1x2 and situated in the first octant is given in the following figure. Find the volume of the solid.

A complex shape in the first octant with height 1, width 5, and length 1. The shape appears to be a slightly deformed quarter of a cylinder of radius 1 and width 5.

The midpoint rule for the triple integral Bf(x,y,z)dV over the rectangular solid box B is a generalization of the midpoint rule for double integrals. The region B is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum li=1mj=1nk=1f(¯xi,¯yj,¯zk)ΔV, where (¯xi,¯yj,¯zk) is the center of the box Bijk and ΔV is the volume of each subbox. Apply the midpoint rule to approximate Bx2dV over the solid B={(x,y,z)|0x1, 0y1, 0z1} by using a partition of eight cubes of equal size. Round your answer to three decimal places.

[Hide Solution]

5160.313

[T]

a. Apply the midpoint rule to approximate Bex2dV over the solid B={(x,y,z)|0x1, 0y1, 0z1} by using a partition of eight cubes of equal size. Round your answer to three decimal places.

b. Use a CAS to improve the above integral approximation in the case of a partition of n3 cubes of equal size, where n=3,4,...,10.

Suppose that the temperature in degrees Celsius at a point (x,y,z) of a solid E bounded by the coordinate planes and x+y+z=5 is T(x,y,z)=xz+5z+10. Find the average temperature over the solid.

[Hide Solution]

352

Suppose that the temperature in degrees Fahrenheit at a point (x,y,z) of a solid E bounded by the coordinate planes and x+y+z=5 is T(x,y,z)=x+y+xy. Find the average temperature over the solid.

Show that the volume of a right square pyramid of height h and side length a is v=ha23 by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length a is 3a332 by using triple integrals.

Show that the volume of a regular right hexagonal pyramid of edge length a is a332 by using triple integrals.

If the charge density at an arbitrary point (x,y,z) of a solid E is given by the function ρ(x,y,z), then the total charge inside the solid is defined as the triple integral Eρ(x,y,z)dV. Assume that the charge density of the solid E enclosed by the paraboloids x=5y2z2 and x=y2+z25 is equal to the distance from an arbitrary point of E to the origin. Set up the integral that gives the total charge inside the solid E.

Contributors

  • OpenStax Calculus 3
  • Apex Calculus
  • Edited by Paul Seeburger (Monroe Community College)

14.4E: Triple Integrals (Exercises 2) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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