5.2E: Mazoezi ya Sehemu ya 5.2

Katika mazoezi ya 1 - 4, onyesha mipaka kama integrals.

1)\(\displaystyle \lim_{n→∞}\sum_{i=1}^n(x^∗_i)Δx\) juu\([1,3]\)

2)\(\displaystyle \lim_{n→∞}\sum_{i=1}^n(5(x^∗_i)^2−3(x^∗_i)^3)Δx\) juu\([0,2]\)

Jibu

\(\displaystyle ∫^2_0(5x^2−3x^3)\,dx\)

3)\(\displaystyle \lim_{n→∞}\sum_{i=1}^n\sin^2(2πx^∗_i)Δx\) juu\([0,1]\)

4)\(\displaystyle \lim_{n→∞}\sum_{i=1}^n\cos^2(2πx^∗_i)Δx\) juu\([0,1]\)

Jibu

\(\displaystyle ∫^1_0\cos^2(2πx)\,dx\)

Katika mazoezi ya 5 - 10, yaliyotolewa\(L_n\) au\(R_n\) kama ilivyoonyeshwa, onyesha mipaka yao\(n→∞\) kama viungo vya uhakika, kutambua vipindi sahihi.

5)\(\displaystyle L_n=\frac{1}{n}\sum_{i=1}^n\frac{i−1}{n}\)

6)\(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n\frac{i}{n}\)

Jibu

\(\displaystyle ∫^1_0x\,dx\)

7)\(\displaystyle Ln=\frac{2}{n}\sum_{i=1}^n(1+2\frac{i−1}{n})\)

8)\(\displaystyle R_n=\frac{3}{n}\sum_{i=1}^n(3+3\frac{i}{n})\)

Jibu

\(\displaystyle ∫^6_3x\,dx\)

9)\(\displaystyle L_n=\frac{2π}{n}\sum_{i=1}^n2π\frac{i−1}{n}\cos(2π\frac{i−1}{n})\)

10\(\displaystyle R_n=\frac{1}{n}\sum_{i=1}^n(1+\frac{i}{n})\log((1+\frac{i}{n})^2)\)

Jibu

\(\displaystyle ∫^2_1x\log(x^2)\,dx\)

Katika mazoezi ya 11 - 16, tathmini ya vipengele vya kazi zilizowekwa kwa kutumia fomu kwa maeneo ya pembetatu na miduara, na uondoe maeneo chini ya\(x\) -axis.

11)

12)

Jibu

\( 1+2⋅2+3⋅3=14\)

13)

14)

Jibu

\(1−4+9=6\)

15)

16)

Jibu

\(1−2π+9=10−2π\)

Katika mazoezi 17 - 24, tathmini muhimu kutumia formula za eneo.

17)\(\displaystyle ∫^3_0(3−x)\,dx\)

18)\(\displaystyle ∫^3_2(3−x)\,dx\)

Jibu

Muhimu ni eneo la pembetatu,\(\frac{1}{2}.\)

19)\(\displaystyle ∫^3_{−3}(3−|x|)\,dx\)

20)\(\displaystyle ∫^6_0(3−|x−3|)\,dx\)

Jibu

Muhimu ni eneo la pembetatu,\(9.\)

21)\(\displaystyle ∫^2_{−2}\sqrt{4−x^2}\,dx\)

22)\(\displaystyle ∫^5_1\sqrt{4−(x−3)^2}\,dx\)

Jibu

Muhimu ni eneo\(\frac{1}{2}πr^2=2π.\)

23)\(\displaystyle ∫^{12}_0\sqrt{36−(x−6)^2}\,dx\)

24)\(\displaystyle ∫^3_{−2}(3−|x|)\,dx\)

Jibu

Muhimu ni eneo la pembetatu “kubwa” chini ya pembetatu “kukosa”,\(9−\frac{1}{2}.\)

Katika mazoezi ya 25 - 28, tumia wastani wa maadili upande wa kushoto (L) na wa kulia (R) ili kuhesabu integrals ya kazi za mstari wa kipande na grafu zinazopitia orodha iliyotolewa ya pointi juu ya vipindi vilivyoonyeshwa.

25)\( {(0,0),(2,1),(4,3),(5,0),(6,0),(8,3)}\) juu\( [0,8]\)

26)\({(0,2),(1,0),(3,5),(5,5),(6,2),(8,0)}\) juu\([0,8]\)

Jibu

\( L=2+0+10+5+4=21,\; R=0+10+10+2+0=22,\; \dfrac{L+R}{2}=21.5\)

27)\( {(−4,−4),(−2,0),(0,−2),(3,3),(4,3)}\) juu\( [−4,4]\)

28)\( {(−4,0),(−2,2),(0,0),(1,2),(3,2),(4,0)}\) juu\( [−4,4]\)

Jibu

\( L=0+4+0+4+2=10,\;R=4+0+2+4+0=10,\;\dfrac{L+R}{2}=10\)

Tuseme kwamba\(\displaystyle ∫^4_0f(x)\,dx=5\) na\(\displaystyle ∫^2_0f(x)\,dx=−3\), na\(\displaystyle ∫^4_0g(x)\,dx=−1\) na\(\displaystyle ∫^2_0g(x)\,dx=2\). Katika mazoezi 29 - 34, compute integrals.

29)\(\displaystyle ∫^4_0(f(x)+g(x))\,dx\)

30)\(\displaystyle ∫^4_2(f(x)+g(x))\,dx\)

Jibu

\(\displaystyle ∫^4_2f(x)\,dx+∫^4_2g(x)\,dx=8−3=5\)

31)\(\displaystyle ∫^2_0(f(x)−g(x))\,dx\)

32)\(\displaystyle ∫^4_2(f(x)−g(x))\,dx\)

Jibu

\(\displaystyle ∫^4_2f(x)\,dx−∫^4_2g(x)\,dx=8+3=11\)

33)\(\displaystyle ∫^2_0(3f(x)−4g(x))\,dx\)

34)\(\displaystyle ∫^4_2(4f(x)−3g(x))\,dx\)

Jibu

\(\displaystyle 4∫^4_2f(x)\,dx−3∫^4_2g(x)\,dx=32+9=41\)

Katika mazoezi 35 - 38, tumia utambulisho wa\(\displaystyle ∫^A_{−A}f(x)\,dx=∫^0_{−A}f(x)\,dx+∫^A_0f(x)\,dx\) kuhesabu integrals.

35)\(\displaystyle ∫^π_{−π}\frac{\sin t}{1+t^2}dt\) (Kidokezo:\(\displaystyle \sin(−t)=−\sin(t))\)

36)\(\displaystyle ∫^{\sqrt{π}}_\sqrt{−π}\frac{t}{1+\cos t}dt\)

Jibu

Integrand ni isiyo ya kawaida; muhimu ni sifuri.

37)\(\displaystyle ∫^3_1(2−x)\,dx\) (kidokezo: Angalia grafu ya\(f\).)

38)\(\displaystyle ∫^4_2(x−3)^3\,dx\) (kidokezo: Angalia grafu ya\(f\).)

Jibu

Integrand ni antisymmetric kwa heshima\(x=3.\) na muhimu ni sifuri.

Katika mazoezi 39 - 44, kutokana\(\displaystyle ∫^1_0x\,dx=\frac{1}{2},\;∫^1_0x^2\,dx=\frac{1}{3},\) na kwamba na\(\displaystyle ∫^1_0x^3\,dx=\frac{1}{4}\), compute integrals.

39)\(\displaystyle ∫^1_0(1+x+x^2+x^3)\,dx\)

40)\(\displaystyle ∫^1_0(1−x+x^2−x^3)\,dx\)

Jibu

\(\displaystyle 1−\frac{1}{2}+\frac{1}{3}−\frac{1}{4}=\frac{7}{12}\)

41)\(\displaystyle ∫^1_0(1−x)^2\,dx\)

42)\(\displaystyle ∫^1_0(1−2x)^3\,dx\)

Jibu

\(\displaystyle ∫^1_0(1−6x+12x^2−8x^3)\,dx=1−6\left( \frac{1}{2} \right)+12\left(\frac{1}{3}\right)−8\left(\frac{1}{4}\right)=1-3+4-2=0\)

43)\(\displaystyle ∫^1_0\left(6x−\tfrac{4}{3}x^2\right)\,dx\)

44)\(\displaystyle ∫^1_0(7−5x^3)\,dx\)

Jibu

\(7−\frac{5}{4}=\frac{23}{4}\)

Katika mazoezi 45 - 50, tumia theorem ya kulinganisha.

45) Onyesha kwamba\(\displaystyle ∫^3_0(x^2−6x+9)\,dx≥0.\)

46) Onyesha kwamba\(\displaystyle ∫^3_{−2}(x−3)(x+2)\,dx≤0.\)

Jibu

Integrand ni hasi juu\([−2,3].\)

47) Onyesha kwamba\(\displaystyle ∫^1_0\sqrt{1+x^3}\,dx≤∫^1_0\sqrt{1+x^2}\,dx\).

48) Onyesha kwamba\(\displaystyle ∫^2_1\sqrt{1+x}\,dx≤∫^2_1\sqrt{1+x^2}\,dx.\)

Jibu

\(x≤x^2\)juu\([1,2]\), hivyo\(\sqrt{1+x}≤\sqrt{1+x^2}\) juu ya\([1,2].\)

49) Onyesha kwamba\(\displaystyle ∫^{π/2}_0\sin tdt≥\frac{π}{4}\) (kidokezo:\(\sin t≥\frac{2t}{π}\) zaidi\( [0,\frac{π}{2}])\)

50) Onyesha hilo\(\displaystyle ∫^{π/4}_{−π/4}\cos t\,dt≥π\sqrt{2}/4\).

Jibu

\(\cos(t)≥\dfrac{\sqrt{2}}{2}\). Kuzidisha kwa urefu wa muda ili kupata usawa.

Katika mazoezi 51 - 56, kupata thamani ya wastani\(f_{ave}\) ya\(f\) kati\(a\) na\(b\), na kupata uhakika\(c\), wapi\(f(c)=f_{ave}\)

51)\( f(x)=x^2,\; a=−1,\; b=1\)

52)\( f(x)=x^5,\; a=−1,\; b=1\)

Jibu

\(f_{ave}=0;\; c=0\)

53)\( f(x)=\sqrt{4−x^2},\; a=0,\; b=2\)

54)\(f(x)=3−|x|,\; a=−3,\; b=3\)

Jibu

\(\frac{3}{2}\)lini\(c=±\frac{3}{2}\)

55)\(f(x)=\sin x,\; a=0,\; b=2π\)

56)\( f(x)=\cos x,\; a=0,\; b=2π\)

Jibu

\(f_{ave}=0;\; c=\dfrac{π}{2},\; \dfrac{3π}{2}\)

Katika mazoezi 57 - 60, takriban thamani ya wastani kwa kutumia kiasi cha Riemann\(L_{100}\) na\(R_{100}\). Jibu lako linalinganishaje na jibu halisi lililopewa?

57) [T]\(y=\ln(x)\) juu ya muda\( [1,4]\); ufumbuzi halisi ni\(\dfrac{\ln(256)}{3}−1.\)

58) [T]\(y=e^{x/2}\) juu ya muda\([0,1]\); ufumbuzi halisi ni\( 2(\sqrt{e}−1).\)

Jibu

\(L_{100}=1.294,\; R_{100}=1.301;\)wastani halisi ni kati ya maadili haya.

59) [T]\(y=\tan x\) juu ya muda\([0,\frac{π}{4}]\); ufumbuzi halisi ni\(\dfrac{2\ln(2)}{π}\).

60) [T]\(y=\dfrac{x+1}{\sqrt{4−x^2}}\) juu ya muda\([−1,1]\); ufumbuzi halisi ni\(\dfrac{π}{6}\).

Jibu

\(L_{100}×(\dfrac{1}{2})=0.5178,\; R_{100}×(\dfrac{1}{2})=0.5294\)

Katika mazoezi 61 - 64, compute thamani ya wastani kwa kutumia jumla ya Riemann\(L_N\) ya kushoto\(N=1,10,100\). Je, usahihi unalinganishaje na thamani halisi iliyotolewa?

61) [T]\(y=x^2−4\) juu ya muda\([0,2]\); ufumbuzi halisi ni\(−\frac{8}{3}\).

62) [T]\(y=xe^{x^2}\) juu ya muda\([0,2]\); ufumbuzi halisi ni\(\frac{1}{4}(e^4−1).\)

Jibu

\(L_1=0,\; L_{10}×(\frac{1}{2})=8.743493,\; L_{100}×(\frac{1}{2})=12.861728.\)Jibu halisi\(≈26.799,\) hivyo\(L_{100}\) si sahihi.

63) [T]\(y=\left(\dfrac{1}{2}\right)^x\) juu ya muda\([0,4]\); ufumbuzi halisi ni\(\dfrac{15}{64\ln(2)}\).

64) [T]\( y=x\sin(x^2)\) juu ya muda\( [−π,0]\); ufumbuzi halisi ni\( \dfrac{\cos(π^2)−1}{2π.}\)

Jibu

\(L_1×(\frac{1}{π})=1.352,L_{10}×(\frac{1}{π})=−0.1837,L_{100}×(1π)=−0.2956.\)Jibu halisi\(≈−0.303,\) hivyo\(L_{100}\) si sahihi kwa decimal ya kwanza.

65) Tuseme kwamba\(\displaystyle A=∫^{2π}_0\sin^2t\,dt\) na\(\displaystyle B=∫^{2π}_0\cos^2t\,dt.\) Onyesha kwamba\(A+B=2π\) na\(A=B.\)

66) Tuseme kwamba\(\displaystyle A=∫^{π/4}_{−π/4}\sec^2 t\,dt=π\) na\(\displaystyle B=∫^{π/4}_{−π/}4\tan^2 t\,dt.\) Onyesha hilo\(A−B=\dfrac{π}{2}\).

Jibu

Tumia\(\tan^2 θ+1=\sec^2 θ.\) Kisha,\(\displaystyle B−A=∫^{π/4}_{−π/4}1\,dx=\frac{π}{2}.\)

67) Onyesha kwamba thamani ya wastani ya\(\sin^2 t\) juu\([0,2π]\) ni sawa na\(1/2.\) Bila hesabu zaidi, onyesha kama thamani ya wastani ya\(\sin^2 t\) juu pia\([0,π]\) ni sawa na\(1/2.\)

68) Onyesha kwamba thamani ya wastani ya\(\cos^2 t\) juu\([0,2π]\) ni sawa na\(1/2.\) Bila hesabu zaidi, onyesha kama thamani ya wastani ya\(\cos^2(t)\) juu pia\([0,π]\) ni sawa na\(1/2.\)

Jibu

\(\displaystyle ∫^{2π}_0\cos^2t\,dt=π,\)hivyo ugawanye\(2π\) na urefu wa muda. \(\cos^2t\)ina kipindi\(π\), hivyo ndiyo, ni kweli.

69) Eleza kwa nini grafu za kazi ya quadratic (parabola)\(p(x)\) na kazi ya mstari\(ℓ(x)\) inaweza kuingiliana katika pointi mbili zaidi. Tuseme kwamba\(p(a)=ℓ(a)\) na\(p(b)=ℓ(b)\), na kwamba\(\displaystyle ∫^b_ap(t)\,dt>∫^b_aℓ(t)dt\). Eleza kwa nini\(\displaystyle ∫^d_cp(t)>∫^d_cℓ(t)\,dt\) wakati wowote\( a≤c<d≤b.\)

70) Tuseme kwamba parabola\(p(x)=ax^2+bx+c\) kufungua chini\((a<0)\) na ina kipeo cha\(y=\dfrac{−b}{2a}>0\). Kwa muda gani\([A,B]\) ni kubwa\(\displaystyle ∫^B_A(ax^2+bx+c)\,dx\) iwezekanavyo?

Jibu

Muhimu ni maximized wakati mtu anatumia muda mkubwa zaidi ambayo\(p\) ni nonnegative. Hivyo,\(A=\frac{−b−\sqrt{b^2−4ac}}{2a}\) na\(B=\frac{−b+\sqrt{b^2−4ac}}{2a}.\)

71) Tuseme\([a,b]\) inaweza kugawanywa katika vipindi\(a=a_0<a_1<a_2<⋯<a_N=b\) vile kwamba ama\(f≥0\) juu\([a_{i−1},a_i]\) au\(f≤0\) juu\([a_{i−1},a_i]\). Weka\(\displaystyle A_i=∫^{a_i}_{a_{i−1}}f(t)\,dt.\)

a. kueleza kwa nini\(\displaystyle ∫^b_af(t)\,dt=A_1+A_2+⋯+A_N.\)

b Kisha, kueleza kwa nini\(\displaystyle ∫^b_af(t)\,dt≤∫^b_a|f(t)|\,dt.\)

72) Tuseme\(f\) na\(g\) ni kazi ya kuendelea kama kwamba\(\displaystyle ∫^d_cf(t)\,dt≤∫^d_cg(t)\,dt\) kwa kila subinterval\([c,d]\) ya\([a,b]\). Eleza\( f(x)≤g(x)\) kwa nini maadili yote ya\(x.\)

Jibu

Ikiwa\(f(t_0)>g(t_0)\) kwa baadhi\(t_0∈[a,b]\), basi tangu\(f−g\) inaendelea, kuna muda unao na vile\(t_0\) vile\( f(t)>g(t)\) zaidi ya muda\([c,d]\), na kisha\(\displaystyle ∫^d_df(t)\,dt>∫^d_cg(t)\,dt\) juu ya kipindi hiki.

73) Tuseme thamani ya wastani ya\(f\) juu\([a,b]\) ni\(1\) na thamani ya wastani ya\(f\) juu\([b,c]\) ni\(1\) wapi\(a<c<b\). Onyesha kuwa thamani ya wastani wa\(f\) zaidi\([a,c]\) ya pia\(1.\)

74) Tuseme kwamba\([a,b]\) inaweza kugawanywa. kuchukua\(a=a_0<a_1<⋯<a_N=b\) vile kwamba thamani ya wastani ya\(f\) juu ya kila subinterval\([a_{i−1},a_i]=1\) ni sawa na 1 kwa kila\( i=1,…,N\). Eleza kwa nini thamani ya wastani ya f juu pia\( [a,b]\) ni sawa na\(1.\)

Jibu

Muhimu wa f juu ya muda ni sawa na muhimu ya wastani wa f juu ya muda huo. Hivyo,\(\displaystyle ∫^b_af(t)\,dt=∫^{a_1}_{a_0}f(t)\,dt+∫^{a_2}_a{1_f}(t)\,dt+⋯+∫^{a_N}_{a_{N+1}}f(t)\,dt=∫^{a_1}_{a_0}1\,dt+∫^{a_2}_{a_1}1\,dt+⋯+∫^{a_N}_{a_{N+1}}1\,dt\)
\( =(a_1−a_0)+(a_2−a_1)+⋯+(a_N−a_{N−1})=a_N−a_0=b−a\).
Kugawanyika kwa njia ya\(b−a\) anatoa utambulisho taka.

75) Tuseme kwamba kwa\(i\) kila\( 1≤i≤N\) mtu anaye\(\displaystyle ∫^i_{i−1}f(t)\,dt=i\). Onyesha kwamba\(\displaystyle ∫^N_0f(t)\,dt=\frac{N(N+1)}{2}.\)

76) Tuseme kwamba kwa\(i\) kila\(1≤i≤N\) mtu anaye\(\displaystyle ∫^i_{i−1}f(t)\,dt=i^2\). Onyesha hilo\(\displaystyle ∫^N_0f(t)\,dt=\frac{N(N+1)(2N+1)}{6}\).

Jibu

\(\displaystyle ∫^N_0f(t)\,dt=\sum_{i=1}^N∫^i_{i−1}f(t)\,dt=\sum_{i=1}^Ni^2=\frac{N(N+1)(2N+1)}{6}\)

77) [T] Compute kushoto na kulia Riemann kiasi\(\displaystyle L_{10}\) na\(R_{10}\) na wastani wao\(\dfrac{L_{10}+R_{10}}{2}\) kwa\( f(t)=t^2\) zaidi ya\( [0,1]\). Kutokana na kwamba\(\displaystyle ∫^1_0t^2\,dt=1/3\), ni sehemu ngapi za decimal ni\( \dfrac{L_{10}+R_{10}}{2}\) sahihi?

78) [T] Compute kushoto na kulia Riemann kiasi,\(L_10\) na\(R_{10}\), na wastani wao\(\dfrac{L_{10}+R_{10}}{2}\) kwa\( f(t)=(4−t^2)\) zaidi ya\([1,2]\). Kutokana na kwamba\(\displaystyle ∫^2_1(4−t^2)\,dt=1.66\), ni sehemu ngapi za decimal ni\(\dfrac{L_{10}+R_{10}}{2}\) sahihi?

Jibu

\( L_{10}=1.815,\;R_{10}=1.515,\;\frac{L_{10}+R_{10}}{2}=1.665,\)hivyo makadirio ni sahihi kwa maeneo mawili decimal.

79) Kama ni\(\displaystyle ∫^5_1\sqrt{1+t^4}\,dt=41.7133...,\) nini\(\displaystyle ∫^5_1\sqrt{1+u^4}\,du?\)

80) Tathmini ya\(\displaystyle ∫^1_0t\,dt\) kutumia kiasi cha mwisho cha kushoto na cha kulia, kila mmoja na mstatili mmoja. Je, wastani wa jumla hizi za kushoto na za kulia zinalinganisha na thamani halisi\(\displaystyle ∫^1_0t\,dt?\)

Jibu

Wastani ni\(1/2,\) ambayo ni sawa na muhimu katika kesi hii.

81) Tathmini\(\displaystyle ∫^1_0t\,dt\) kwa kulinganisha na eneo la mstatili mmoja na urefu sawa na thamani ya\(t\) katikati\(t=\dfrac{1}{2}\). Je, makadirio haya ya midpoint yanalinganishaje na thamani halisi?\(\displaystyle ∫^1_0t\,dt?\)

82) Kutoka kwenye grafu ya\(\sin(2πx)\) umeonyeshwa:

a. kueleza kwa nini\(\displaystyle ∫^1_0\sin(2πt)\,dt=0.\)

b Eleza kwa nini, kwa ujumla,\(\displaystyle ∫^{a+1}_a\sin(2πt)\,dt=0\) kwa thamani yoyote ya\(a\).

Jibu

a. grafu ni antisymmetric kwa heshima na\(t=\frac{1}{2}\) zaidi\([0,1]\), hivyo thamani ya wastani ni sifuri.
b Kwa thamani yoyote ya\(a\), grafu kati\([a,a+1]\) ni mabadiliko ya grafu juu\([0,1]\), hivyo maeneo ya wavu juu na chini ya mhimili hazibadilika na wastani hubakia sifuri.

83) Kama f ni 1-mara kwa mara\((f(t+1)=f(t))\), isiyo ya kawaida, na integrable juu ya\([0,1]\), ni daima kweli kwamba\(\displaystyle ∫^1_0f(t)\,dt=0?\)

84) Ikiwa f ni 1-mara kwa mara na\(\displaystyle ∫10f(t)\,dt=A,\) ni lazima kweli kwamba\(\displaystyle ∫^{1+a}_af(t)\,dt=A\) kwa wote\(A\)?

Jibu

Ndiyo, muhimu juu ya muda wowote wa urefu 1 ni sawa.