ਪੜਾਅ
1212 ਇਕਾਈਆਂ ਵਰਗ
310310 ਇਕਾਈਆਂ ਵਰਗ
2+222+22 ਇਕਾਈਆਂ ਵਰਗ
5353 ਇਕਾਈਆਂ ਵਰਗ
5353 ਇਕਾਈਆਂ ਵਰਗ
ਪਾਈ 2 ਪਾਈ 2
8ਪਾਈ8ਪਾਈ ਘਣ ਇਕਾਈਆਂ
21ਪਾਈ21ਪਾਈ ਘਣ ਇਕਾਈਆਂ
10ਪਾਈ310ਪਾਈ3 ਘਣ ਇਕਾਈਆਂ
60ਪਾਈ60ਪਾਈ ਘਣ ਇਕਾਈਆਂ
15ਪਾਈ215ਪਾਈ2 ਘਣ ਇਕਾਈਆਂ
8ਪਾਈ8ਪਾਈ ਘਣ ਇਕਾਈਆਂ
12ਪਾਈ12ਪਾਈ ਘਣ ਇਕਾਈਆਂ
11ਪਾਈ611ਪਾਈ6 ਘਣ ਇਕਾਈਆਂ
ਪਾਈ6ਪਾਈ6 ਘਣ ਇਕਾਈਆਂ
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
ਕਮਾਨ ਦੀ ਲੰਬਾਈ ≈ 3.8202 ਕਮਾਨ ਦੀ ਲੰਬਾਈ ≈ 3.8202
ਕਮਾਨ ਦੀ ਲੰਬਾਈ = 3.15018 ਕਮਾਨ ਦੀ ਲੰਬਾਈ = 3.15018
ਪਾਈ 6 ( 5 5 − 3 3 ) ≈ 3.133 ਪਾਈ 6 ( 5 5 − 3 3 ) ≈ 3.133
12 ਪਾਈ 12 ਪਾਈ
70 / 3 70 / 3
24 ਪਾਈ 24 ਪਾਈ
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 2.1 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 ਕੁੱਲ ਮੁਨਾਫ਼ਾ 200 ਸੈੱਲ ਫੋਨਾਂ ਦੀ ਵਿਕਰੀ ਲਈ
3.323.32 ਮੀਲ ਇਹ ਦਰਸਾਉਂਦਾ ਹੈ ਕਿ ਖਰਗੋਸ਼ ਕੱਛੂ ਤੋਂ ਕਿੰਨਾ ਅੱਗੇ ਹੈ
343 24 343 24
4 3 4 3
π − 32 25 π − 32 25
ਦਫ਼ਾ 2.2 ਕਸਰਤਾਂ
8 ਇਕਾਈਆਂ³
32323232 ਇਕਾਈਆਂ³
724πr2h724πr2h ਇਕਾਈਆਂ³
π24π24 ਇਕਾਈਆਂ³
22 ਇਕਾਈਆਂ³
π240π240 ਇਕਾਈਆਂ³
4096π54096π5 ਇਕਾਈਆਂ³
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 2.3 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 2.4 ਅਭਿਆਸ
2 26 2 26
2 17 2 17
π 6 ( 17 17 − 5 5 ) π 6 ( 17 17 − 5 5 )
13 13 − 8 27 13 13 − 8 27
4 3 4 3
2.0035 2.0035
123 32 123 32
10 10
20 3 20 3
1 675 ( 229 229 − 8 ) 1 675 ( 229 229 − 8 )
1 8 ( 4 5 + ln ( 9 + 4 5 ) ) 1 8 ( 4 5 + ln ( 9 + 4 5 ) )
1.201 1.201
15.2341 15.2341
49 π 3 49 π 3
70 π 2 70 π 2
8 π 8 π
120 π 26 120 π 26
π 6 ( 17 17 − 1 ) π 6 ( 17 17 − 1 )
9 2 π 9 2 π
10 10 π 27 ( 73 73 − 1 ) 10 10 π 27 ( 73 73 − 1 )
25.645 25.645
π ( 2 + π ) π ( 2 + π )
10.5017 10.5017
2424 ft
2 2
Answers may vary
1. ਗੈਬਰੀਅਲ ਦੇ ਸਿੰਗ ਬਾਰੇ ਹੋਰ ਜਾਣਕਾਰੀ ਲਈ ਦੇਖੋ।
2. ਭਾਗ 2.5 ਅਭਿਆਸ
3. 150150 ਫੁੱਟ-ਪੌਂਡ
4. 200 ਜੇ 200 ਜੇ
5. 11 ਜੇ
6. 39 2 39 2
7. ln ( 243 ) ln ( 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 ਸੈ.ਮੀ.
13. 3636 ਜੇ
14. 18,75018,750 ਫੁੱਟ-ਪੌਂਡ
15. ਭਾਰ = 323 × 10⁹ ਪੌਂਡ = 323 × 10⁹ ਪੌਂਡ
16. ਕੰਮ = 43 × 10¹² ਫੁੱਟ-ਪੌਂਡ = 43 × 10¹² ਫੁੱਟ-ਪੌਂਡ
17. 9.71 × 10⁸ N m 9.71 × 10⁸ N m
18. ਏ. 3,000,0003,000,000 ਪੌਂਡ, ਬੀ. 750,000750,000 ਪੌਂਡ
19. 23.25π23.25π ਮਿਲੀਅਨ ਫੁੱਟ-ਪੌਂਡ
20. A ρ H ² 2 A ρ H ² 2
21. ਜਵਾਬ ਵੱਖੋ-ਵੱਖਰੇ ਹੋ ਸਕਦੇ ਹਨ
22. ਭਾਗ 2.6 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 2.7 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 2.8 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 2.9 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 ) )