ਚੌਕੀ
1212 ਇਕਾਈਆਂ ਵਰਗ
310310 ਇਕਾਈਆਂ ਵਰਗ
2+222+22 ਇਕਾਈਆਂ ਵਰਗ
5353 ਇਕਾਈਆਂ ਵਰਗ
5353 ਇਕਾਈਆਂ ਵਰਗ
π 2 π 2
8π8π ਇਕਾਈਆਂ ਘਣ
21π21π ਇਕਾਈਆਂ ਘਣ
10π310π3 ਇਕਾਈਆਂ ਘਣ
60π60π ਇਕਾਈਆਂ ਘਣ
15π215π2 ਇਕਾਈਆਂ ਘਣ
8π8π ਇਕਾਈਆਂ ਘਣ
12π12π ਇਕਾਈਆਂ ਘਣ
11π611π6 ਇਕਾਈਆਂ ਘਣ
π6π6 ਇਕਾਈਆਂ ਘਣ
ਵਾਸ਼ਰਾਂ ਦੀ ਵਿਧੀ ਵਰਤੋ; V=∫−11π[(2−x2)2−(x2)2]dxV=∫−11π[(2−x2)2−(x2)2]dx
1 6 ( 5 5 − 1 ) ≈ 1.697 1 6 ( 5 5 − 1 ) ≈ 1.697
ਚਾਪ ਲੰਬਾਈ ≈ 3.8202 ਚਾਪ ਲੰਬਾਈ ≈ 3.8202
ਚਾਪ ਲੰਬਾਈ = 3.15018 ਚਾਪ ਲੰਬਾਈ = 3.15018
π 6 ( 5 5 − 3 3 ) ≈ 3.133 π 6 ( 5 5 − 3 3 ) ≈ 3.133
12 π 12 π
70 / 3 70 / 3
24 π 24 π
88 ft-lb
Approximately 43,255.243,255.2 ft-lb
156,800156,800 N
Approximately 7,164,520,000 lb or 3,582,260 t
M = 24 , x – = 2 5 m M = 24 , x – = 2 5 m
(−1,−1)(−1,−1) m
The centroid of the region is (3/2,6/5).(3/2,6/5).
The centroid of the region is (1,13/5).(1,13/5).
The centroid of the region is (0,2/5).(0,2/5).
6π26π2 units3
ddxln(2x2+x)=4x+12x2+xddxln(2x2+x)=4x+12x2+x
ddx(ln(x3))2=6ln(x3)xddx(ln(x3))2=6ln(x3)x
∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C ∫ x 2 x 3 + 6 d x = 1 3 ln | x 3 + 6 | + C
4 ln 2 4 ln 2
ddx(ex2e5x)=ex2−5x(2x−5)ddx(ex2e5x)=ex2−5x(2x−5)
ddt(e2t)3=6e6tddt(e2t)3=6e6t
∫ 4 e 3 x d x = − 4 3 e −3 x + C ∫ 4 e 3 x d x = − 4 3 e −3 x + C
ddt4t4=4t4(ln4)(4t3)ddt4t4=4t4(ln4)(4t3)
ddxlog3(x2+1)=x(ln3)(x2+1)ddxlog3(x2+1)=x(ln3)(x2+1)
∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C ∫ x 2 2 x 3 d x = 1 3 ln 2 2 x 3 + C
There are 81,377,39681,377,396 bacteria in the population after 44 hours. The population reaches 100100 million bacteria after 244.12244.12 minutes.
At 5%5% interest, she must invest $223,130.16.$223,130.16. At 6%6% interest, she must invest $165,298.89.$165,298.89.
38.9038.90 months
The coffee is first cool enough to serve about 3.53.5 minutes after it is poured. The coffee is too cold to serve about 77 minutes after it is poured.
A total of 94.1394.13 g of carbon remains. The artifact is approximately 13,30013,300 years old.
ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)
ddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshxddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshx
∫sinh3xcoshxdx=sinh4x4+C∫sinh3xcoshxdx=sinh4x4+C
∫sech2(3x)dx=tanh(3x)3+C∫sech2(3x)dx=tanh(3x)3+C
ddx(cosh−1(3x))=39x2−1ddx(cosh−1(3x))=39x2−1
ddx(coth−1x)3=3(coth−1x)21−x2ddx(coth−1x)3=3(coth−1x)21−x2
∫1x2−4dx=cosh−1(x2)+C∫1x2−4dx=cosh−1(x2)+C
∫11−e2xdx=−sech−1(ex)+C∫11−e2xdx=−sech−1(ex)+C
52.95 ft 52.95 ft
Section 6.1 Exercises
32 3 32 3
13 12 13 12
36 36
243 square units
4
2(e−1)2e2(e−1)2e
1313
343343
5252
1212
9292
9292
332332
e−2e−2
274274
43−ln(3)43−ln(3)
1212
1212
−2(2−π)−2(2−π)
1.067 1.067
1.605 1.605
7.523 7.523
3 π − 4 12 3 π − 4 12
1.429 1.429
$33,333.33$33,333.33 total profit for 200200 cell phones sold
3.323.32 mi represents how far ahead the hare is from the tortoise
343 24 343 24
4 3 4 3
π − 32 25 π − 32 25
Section 6.2 Exercises
8 units3
32323232 units3
724πr2h724πr2h units3
π24π24 units3
22 units3
π240π240 units3
4096π54096π5 units3
8π98π9 units3
π2π2 units3
207π207π units3
4π54π5 units3
16π316π3 units3
ππ units3
16π316π3 units3
72π572π5 units3
108π5108π5 units3
3π103π10 units3
26π26π units3
9π9π units3
π20(75−4ln5(2))π20(75−4ln5(2)) units3
m2π3(b3−a3)m2π3(b3−a3) units3
4a2bπ34a2bπ3 units3
2π22π2 units3
2ab2π32ab2π3 units3
V=2π3(2r+h)r-h2V=2π3(2r+h)r-h2 units3
π3(h+R)(h−2R)2π3(h+R)(h−2R)2 units3
Section 6.3 Exercises
54π54π units3
81π81π units3
512π7512π7 units3
2π2π units3
2π32π3 ਘਣ ਇਕਾਈਆਂ
2π2π ਘਣ ਇਕਾਈਆਂ
4π54π5 ਘਣ ਇਕਾਈਆਂ
64π364π3 ਘਣ ਇਕਾਈਆਂ
32π532π5 ਘਣ ਇਕਾਈਆਂ
7 π 6
48 π
308 π 5
512 π 7
8 π 5
28π1528π15 ਘਣ ਇਕਾਈਆਂ
3π103π10 ਘਣ ਇਕਾਈਆਂ
π6·22/35-1110=π1012·22/3-11≈2.5286π6·22/35-1110=π1012·22/3-11≈2.5286 ਘਣ ਇਕਾਈਆਂ
0.98760.9876 ਘਣ ਇਕਾਈਆਂ
3232 ਘਣ ਇਕਾਈਆਂ
496π15496π15 ਘਣ ਇਕਾਈਆਂ
398π15398π15 ਘਣ ਇਕਾਈਆਂ
15.907415.9074 ਘਣ ਇਕਾਈਆਂ
13πr2h13πr2h ਘਣ ਇਕਾਈਆਂ
πr2hπr2h ਘਣ ਇਕਾਈਆਂ
πa2πa2 ਘਣ ਇਕਾਈਆਂ
ਭਾਗ 6.4 ਅਭਿਆਸ
2 26 2 26
2 17 2 17
1. π 6 ( 17 17 − 5 5 ) π 6 ( 17 17 − 5 5 )
2. 13 13 − 8 27 13 13 − 8 27
3. 4 3 4 3
4. 2.0035 2.0035
5. 123 32 123 32
6. 10 10
7. 20 3 20 3
8. 1 675 ( 229 229 − 8 ) 1 675 ( 229 229 − 8 )
9. 1 8 ( 4 5 + ln ( 9 + 4 5 ) ) 1 8 ( 4 5 + ln ( 9 + 4 5 ) )
10. 1.201 1.201
11. 15.2341 15.2341
12. 49 π 3 49 π 3
13. 70 π 2 70 π 2
14. 8 π 8 π
15. 120 π 26 120 π 26
16. π 6 ( 17 17 − 1 ) π 6 ( 17 17 − 1 )
17. 9 2 π 9 2 π
18. 10 10 π 27 ( 73 73 − 1 ) 10 10 π 27 ( 73 73 − 1 )
19. 25.645 25.645
20. π ( 2 + π ) π ( 2 + π )
21. 10.5017 10.5017
22. 2424 ft
23. 2 2
24. ਜਵਾਬ ਵੱਖੋ-ਵੱਖਰੇ ਹੋ ਸਕਦੇ ਹਨ।
1. ਗੈਬਰੀਅਲ ਦੇ ਸਿੰਗ ਬਾਰੇ ਹੋਰ ਜਾਣਕਾਰੀ ਲਈ ਵੇਖੋ।
2. ਭਾਗ 6.5 ਅਭਿਆਸ
3. 150150 ਫੁੱਟ-ਪੌਂਡ
4. 200 ਜੇ 200 ਜੇ
5. 11 ਜੇ
6. 39 2 39 2
7. ਲਘੂਗਣਕ ( 243 ) ਲਘੂਗਣਕ ( 243 )
8. 332 π 15 332 π 15
9. 100 π 100 π
10. 20 π 15 20 π 15
11. 66 ਜੇ
12. 55 ਸੈਂਟੀਮੀਟਰ
13. 3636 ਜੇ
14. 18,75018,750 ਫੁੱਟ-ਪੌਂਡ
15. ਭਾਰ = 323 × 109 ਪੌਂਡ = 323 × 109 ਪੌਂਡ
16. ਕਾਰਜ = 43 × 1012 ਫੁੱਟ-ਪੌਂਡ = 43 × 1012 ਫੁੱਟ-ਪੌਂਡ
17. 9.71 × 10 8 N m 9.71 × 10 8 N m
18. ਏ. 3,000,0003,000,000 ਪੌਂਡ, ਬੀ. 750,000750,000 ਪੌਂਡ
19. 23.25 π 23.25 π ਮਿਲੀਅਨ ਫੁੱਟ-ਪੌਂਡ
20. ਏ ρ ਐਚ 2 2 ਏ ρ ਐਚ 2 2
21. ਜਵਾਬ ਵੱਖ-ਵੱਖ ਹੋ ਸਕਦੇ ਹਨ
22. ਭਾਗ 6.6 ਅਭਿਆਸ
23. 5 4 5 4
24. ( 2 3 , 2 3 ) ( 2 3 , 2 3 )
( 7 4 , 3 2 ) ( 7 4 , 3 2 )
3 L 4 3 L 4
π 2 π 2
e 2 + 1 e 2 − 1 e 2 + 1 e 2 − 1
π 2 − 4 π π 2 − 4 π
1 4 ( 1 + e 2 ) 1 4 ( 1 + e 2 )
( a 3 , b 3 ) ( a 3 , b 3 )
( 0 , π 8 ) ( 0 , π 8 )
( 0 , 3 ) ( 0 , 3 )
( 0 , 4 π ) ( 0 , 4 π )
( 5 8 , 1 3 ) ( 5 8 , 1 3 )
2 m π 3 2 m π 3
π a 2 b π a 2 b
( 4 3 π , 4 3 π ) ( 4 3 π , 4 3 π )
( 1 2 , 2 5 ) ( 1 2 , 2 5 )
( 0 , 28 9 π ) ( 0 , 28 9 π )
Center of mass: (3a8,2a25),(3a8,2a25), volume: πa42πa42
Volume: 2π2a2(b+a)2π2a2(b+a)
Section 6.7 Exercises
1 x 1 x
− 1 x ( ln x ) 2 − 1 x ( ln x ) 2
ln ( x + 1 ) + C ln ( x + 1 ) + C
ln ( x ) + 1 ln ( x ) + 1
cot ( x ) cot ( x )
7 x 7 x
csc ( x ) sec x csc ( x ) sec x
−2 tan x −2 tan x
1 2 ln ( 5 3 ) 1 2 ln ( 5 3 )
2 − 1 2 ln ( 5 ) 2 − 1 2 ln ( 5 )
1 ln ( 2 ) − 1 1 ln ( 2 ) − 1
1 2 ln ( 2 ) 1 2 ln ( 2 )
1 3 ( ln x ) 3 1 3 ( ln x ) 3
2 x 3 x 2 + 1 x 2 − 1 2 x 3 x 2 + 1 x 2 − 1
x −2 − ( 1 / x ) ( ln x − 1 ) x −2 − ( 1 / x ) ( ln x − 1 )
e x e − 1 e x e − 1
1 1
− 1 x 2 − 1 x 2
π − ln ( 2 ) π − ln ( 2 )
1 x 1 x
e 5 − 6 units 2 e 5 − 6 units 2
ln ( 4 ) − 1 units 2 ln ( 4 ) − 1 units 2
2.8656 2.8656
3.1502 3.1502
Section 6.8 Exercises
True
True; k=ln(2)tk=ln(2)t
2020 hours
No. The relic is approximately 871871 years old.
7373 years
55 days 66 hours 2727 minutes
12 12
8.618 % 8.618 %
$ 6766.76 $ 6766.76
99 hours 1313 minutes
239,179239,179 years
P′(t)=43e0.01604t.P′(t)=43e0.01604t. The population is always increasing.
The population reaches 1010 billion people in 2032.2032.
P′(t)=2.259e0.06407t.P′(t)=2.259e0.06407t. The population is always increasing.
Section 6.9 Exercises
e x and e − x e x and e − x
Answers may vary
Answers may vary
Answers may vary
3 sinh ( 3 x + 1 ) 3 sinh ( 3 x + 1 )
− tanh ( x ) sech ( x ) − tanh ( x ) sech ( x )
4 cosh ( x ) sinh ( x ) 4 cosh ( x ) sinh ( x )
x sech 2 ( x 2 + 1 ) x 2 + 1 x sech 2 ( x 2 + 1 ) x 2 + 1
6 sinh 5 ( x ) cosh ( x ) 6 sinh 5 ( x ) cosh ( x )
1 2 sinh ( 2 x + 1 ) + C 1 2 sinh ( 2 x + 1 ) + C
1 2 sinh ( x 2 ) + C 1 2 sinh ( x 2 ) + C
1 3 cosh 3 ( x ) + C 1 3 cosh 3 ( x ) + C
ln ( 1 + cosh ( x ) ) + C ln ( 1 + cosh ( x ) ) + C
cosh ( x ) + sinh ( x ) + C cosh ( x ) + sinh ( x ) + C
4 1 − 16 x 2 4 1 − 16 x 2
sinh ( x ) cosh 2 ( x ) + 1 sinh ( x ) cosh 2 ( x ) + 1
− csc ( x ) − csc ( x )
− 1 ( x 2 − 1 ) tanh −1 ( x ) − 1 ( x 2 − 1 ) tanh −1 ( x )
1 a tanh −1 ( x a ) + C 1 a tanh −1 ( x a ) + C
x 2 + 1 + C x 2 + 1 + C
cosh −1 ( e x ) + C cosh −1 ( e x ) + C
Answers may vary
37.30 37.30
y = 1 c cosh ( c x ) y = 1 c cosh ( c x )
−0.521095 −0.521095
10 10
Review Exercises
False
False
16 3 units 3 16 3 units 3
162 π 5 162 π 5
a. 4,4, b. 128π7,128π7, c. 64π564π5
a. 1.949,1.949, b. 21.952,21.952, c. 17.09917.099
a. 316,316, b. 452π15,452π15, c. 31π631π6
π ln 17 π ln 17
Mass: 12,12, center of mass: (1835,922)(1835,922)
17 + 1 8 ln ( 33 + 8 17 ) 17 + 1 8 ln ( 33 + 8 17 )
Volume: 3π4,3π4, surface area: π(2−sinh−1(1)+sinh−1(16)−25716)π(2−sinh−1(1)+sinh−1(16)−25716)
11:02 a.m.
π ( 1 + sinh ( 1 ) cosh ( 1 ) ) π ( 1 + sinh ( 1 ) cosh ( 1 ) )