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)|0≤x≤1, 0≤y≤2, 0≤z≤3}
- Answer
-
192
2. ∭B(xy+yz+xz) dV, where B={(x,y,z)|1≤x≤2, 0≤y≤2, 1≤z≤3}
3. ∭B(x cos y+z) dV, where B={(x,y,z)|0≤x≤1, 0≤y≤π, −1≤z≤1}
- Answer
-
0
4. ∭B(z sin x+y2) dV, where B={(x,y,z)|0≤x≤π, 0≤y≤1, −1≤z≤2}
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. ∫10∫21∫32(x2+ln y+z) dx dy dz
- Answer
-
∫10∫21∫32(x2+ln y+z) dx dy dz=356+2 ln2
2. ∫10∫1−1∫30(zex+2y) dx dy dz
3. ∫2−1∫31∫40(x2z+1y) dx dy dz
- Answer
-
∫2−1∫31∫40(x2z+1y) dx dy dz=64+12 ln 3
4. ∫21∫−1−2∫10x+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
∫ba∫dc∫feF(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
∫ba∫dc∫feF′(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)|a≤x≤b, h1(x)≤y≤h2(x), e≤z≤f}.
1. ∭E(2x+5y+7z) dV, where E={(x,y,z)|0≤x≤1, 0≤y≤−x+1, 1≤z≤2}
- Answer
-
7712
2. ∭E(y ln x+z) dV, where E={(x,y,z)|1≤x≤e, 0≤yln x, 0≤z≤1}
3. ∭E(sin x+sin y)dV, where E={(x,y,z)|0≤x≤π2, −cos x≤y≤cos x, −1≤z≤1}
- Answer
-
2
4. ∭E(xy+yz+xz)dV where E={(x,y,z)|0≤x≤1, −x2≤y≤x2, 0≤z≤1}
In the following exercises, evaluate the triple integrals over the indicated bounded region E.
5. ∭E(x+2yz) dV, where E={(x,y,z)|0≤x≤1, 0≤y≤x, 0≤z≤5−x−y}
- Answer
-
430120
6. ∭E(x3+y3+z3) dV, where E={(x,y,z)|0≤x≤2, 0≤y≤2x, 0≤z≤4−x−y}
7. ∭Ey dV, where E={(x,y,z)|−1≤x≤1, −√1−x2≤y≤√1−x2, 0≤z≤1−x2−y2}
- Answer
-
0
8. ∭Ex dV, where E={(x,y,z)|−2≤x≤2, −4√1−x2≤y≤√4−x2, 0≤z≤4−x2−y2}
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)≤x≤g2(y), c≤y≤d, e≤z≤f}.
1. ∭Ex2 dV, where E={(x,y,z)|1−y2≤x≤y2−1, −1≤y≤1, 1≤z≤2}
- Answer
-
−64105
2. ∭E(sin x+y) dV, where E={(x,y,z)|−y4≤x≤y4, 0≤y≤2, 0≤z≤4}
3. ∭E(x−yz) dV, where E={(x,y,z)|−y6≤x≤√y, 0≤y≤1, −1≤z≤1}
- Answer
-
1126
4. ∭Ez dV, where E={(x,y,z)|2−2y≤x≤2+√y, 0≤y≤1, 2≤z≤3}
Exercise 7.4E.6
In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)|g1(y)≤x≤g2(y), c≤y≤d, u1(x,y)≤z≤u2(x,y)}
1. ∭Ez dV, where E={(x,y,z)|−y≤x≤y, 0≤y≤1, 0≤z≤1−x4−y4}
- Answer
-
113450
2. ∭E(xz+1) dV, where E={(x,y,z)|0≤x≤√y, 0≤y≤2, 0≤z≤1−x2−y2}
3. ∭E(x−z) dV, where E={(x,y,z)|−√1−y2≤x≤y, 0≤y≤12x, 0≤z≤1−x2−y2}
- Answer
-
1160(6√3−41)
4. ∭E(x+y) dV, where E={(x,y,z)|0≤x≤√1−y2, 0≤y≤1x, 0≤z≤1−x}
In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)|(x,y)∈D, u1(x,y)x≤z≤u2(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+y2≤1}
- Answer
-
3π2
6. ∬D(∫31x(z+1) dz) dA, where D={(x,y)|x2−y2≥1, x≤√5}
7. ∬D(∫10−x−y0(x+2z) dz) dA, where D={(x,y)|y≥0, x≥0, x+y≤10}
- Answer
-
1250
8. ∬D(∫4x2+4y20y dz) dA, where D={(x,y)|x2+y2≤4, y≥1, x≥0}
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.
- Answer
- ∫50∫3−3∫√9−y20zdzdydx=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. ∫10∫31∫42(x2z2+1)dx dy dz
- Answer
-
∫10∫31∫42(y2z2+1)dz dx dy; ∫10∫31∫42(x2z2+1)dx dy dz.
2. ∫30∫10∫−x+10(2x+5y+7z)dy dx dz
3. ∫10∫y−y∫1−x4−y40ln xdz dx dy
4. ∫1−1∫10∫√y−y6(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=∫a−a∫√a2−z2−√a2−z2∫a2√x2+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.