Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

7.4E:

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 7.4E.1

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

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

Answer

192

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

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

Answer

0

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

Exercise 7.4E.2

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

1. 102132(x2+ln y+z) dx dy dz

Answer

102132(x2+ln y+z) dx dy dz=356+2 ln2

2. 101130(zex+2y) dx dy dz

3. 213140(x2z+1y) dx dy dz

Answer

213140(x2z+1y) dx dy dz=64+12 ln 3

4. 211210x+yz dx dy dz

Exercise 7.4E.3

1. 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).

2. 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)].

Exercise 7.4E.4

In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)|axb, h1(x)yh2(x), ezf}.

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

Answer

7712

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

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

Answer

2

4. 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.

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

Answer

430120

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

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

Answer

0

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

Exercise 7.4E.5

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}.

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

Answer

64105

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

3. E(xyz) dV, where E={(x,y,z)|y6xy, 0y1, 1z1}

Answer

1126

4. Ez dV, where E={(x,y,z)|22yx2+y, 0y1, 2z3}

Exercise 7.4E.6

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)}

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

Answer

113450

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

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

Answer

1160(6341)

4. 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

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

Answer

3π2

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

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

Answer

1250

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

Exercise 7.4E.7

1. The solid E bounded by y2+z2=9, z=0, and x=5 is shown in the following figure. Evaluate the integral

2. The solid E bounded by y2+z2=9,z=0, and x=5 is shown in the following figure. Evaluate the integral EzdV 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.

Answer
50339y20zdzdydx=90

 

Exercise 7.4E.8

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.

1. 103142(x2z2+1)dx dy dz

Answer

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

2. 3010x+10(2x+5y+7z)dy dx dz

3. 10yy1x4y40ln xdz dx dy

4. 1110yy6(x+yz)dx dy dz

Exercise 7.4E.9

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

Answer

V=aaa2z2a2z2a2x2+z2dy dx dz

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

Exercise 7.4E.10 Average value

1. 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.

Answer

92

2. 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


This page titled 7.4E: is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

Support Center

How can we help?