11.4E: Mazoezi ya Sehemu ya 11.4

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Edwin “Jed” Herman & Gilbert Strang
OpenStax

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Katika mazoezi ya 1 -13, tambua muhimu ya uhakika ambayo inawakilisha eneo hilo.

1) Mkoa ulioambatanishwa na\(r=4\)

2) Mkoa ulioambatanishwa na\(r=3\sin θ\)

Jibu: \(\displaystyle\frac{9}{2}∫^π_0\sin^2θ\,dθ\)

3) Mkoa katika quadrant ya kwanza ndani ya cardioid\(r=1+\sin θ\)

4) Mkoa iliyoambatanishwa na petal moja ya\(r=8\sin(2θ)\)

Jibu: \(\displaystyle\frac{3}{2}∫^{π/2}_0\sin^2(2θ)\,dθ\)

5) Mkoa iliyoambatanishwa na petal moja ya\(r=cos(3θ)\)

6) Mkoa chini ya mhimili wa polar na iliyofungwa na\(r=1−\sin θ\)

Jibu: \(\displaystyle\frac{1}{2}∫^{2π}_π(1−\sin θ)^2\,dθ\)

7) Mkoa katika roboduara ya kwanza iliyoambatanishwa na\(r=2−\cos θ\)

8) Mkoa iliyoambatanishwa na kitanzi ndani ya\(r=2−3\sin θ\)

Jibu: \(\displaystyle∫^{π/2}_{\sin^{−1}(2/3)}(2−3\sin θ)^2\,dθ\)

9) Mkoa iliyoambatanishwa na kitanzi ndani ya\(r=3−4\cos θ\)

10) Mkoa uliofungwa\(r=1−2\cos θ\) na nje ya kitanzi cha ndani

Jibu: \(\displaystyle∫^π_0(1−2\cos θ)^2\,dθ−∫^{π/3}_0(1−2\cos θ)^2\,dθ\)

11) Mkoa wa kawaida\(r=3\sin θ\) na\(r=2−\sin θ\)

12) Mkoa wa kawaida\(r=2\) na\(r=4\cos θ\)

Jibu: \(\displaystyle4∫^{π/3}_0\,dθ+16∫^{π/2}_{π/3}(\cos^2θ)\,dθ\)

13) Mkoa wa kawaida\(r=3\cos θ\) na\(r=3\sin θ\)

Katika mazoezi 14 -26, tafuta eneo la kanda iliyoelezwa.

14) Iliyoambatanishwa na\(r=6\sin θ\)

Jibu: \(9π\text{ units}^2\)

15) Juu ya mhimili wa polar iliyofungwa na\(r=2+\sin θ\)

16) Chini ya mhimili wa polar na umefungwa na\(r=2−\cos θ\)

Jibu: \(\frac{9π}{4}\text{ units}^2\)

17) Imefungwa na petal moja ya\(r=4\cos(3θ)\)

18) Imefungwa na petal moja ya\(r=3\cos(2θ)\)

Jibu: \(\frac{9π}{8}\text{ units}^2\)

19) Iliyoambatanishwa na\(r=1+\sin θ\)

20) Iliyoambatanishwa na kitanzi ndani ya\(r=3+6\cos θ\)

Jibu: \(\frac{18π−27\sqrt{3}}{2}\text{ units}^2\)

21) Imefungwa\(r=2+4\cos θ\) na nje ya kitanzi cha ndani

22) Mambo ya ndani ya kawaida\(r=4\sin(2θ)\) na\(r=2\)

Jibu: \(\frac{4}{3}(4π−3\sqrt{3})\text{ units}^2\)

23) Mambo ya ndani ya kawaida\(r=3−2\sin θ\) na\(r=−3+2\sin θ\)

24) Mambo ya ndani ya kawaida\(r=6\sin θ\) na\(r=3\)

Jibu: \(\frac{3}{2}(4π−3\sqrt{3})\text{ units}^2\)

25) Ndani\(r=1+\cos θ\) na nje\(r=\cos θ\)

26) Mambo ya ndani ya kawaida\(r=2+2\cos θ\) na\(r=2\sin θ\)

Jibu: \((2π−4)\text{ units}^2\)

Katika mazoezi 27 - 30, tafuta muhimu ya uhakika ambayo inawakilisha urefu wa arc.

27)\(r=4\cos θ\) kwa muda\(0≤θ≤\frac{π}{2}\)

28)\(r=1+\sin θ\) kwa muda\(0≤θ≤2π\)

Jibu: \(\displaystyle∫^{2π}_0\sqrt{(1+\sin θ)^2+\cos^2θ}\,dθ\)

29)\(r=2\sec θ\) kwa muda\(0≤θ≤\frac{π}{3}\)

30)\(r=e^θ\) kwa muda\(0≤θ≤1\)

Jibu: \(\displaystyle\sqrt{2}∫^1_0e^θ\,dθ\)

Katika mazoezi 31 - 35, pata urefu wa curve juu ya muda uliopewa.

31)\(r=6\) kwa muda\(0≤θ≤\frac{π}{2}\)

32)\(r=e^{3θ}\) kwa muda\(0≤θ≤2\)

Jibu: \(\frac{\sqrt{10}}{3}(e^6−1)\)vitengo

33)\(r=6\cos θ\) kwa muda\(0≤θ≤\frac{π}{2}\)

34)\(r=8+8\cos θ\) kwa muda\(0≤θ≤π\)

Jibu: \(32\)vitengo

35)\(r=1−\sin θ\) kwa muda\(0≤θ≤2π\)

Katika mazoezi 36 - 40, tumia uwezo wa ushirikiano wa calculator ili takriban urefu wa curve.

36) [T]\(r=3θ\) juu ya muda\(0≤θ≤\frac{π}{2}\)

Jibu: \(6.238\)vitengo

37) [T]\(r=\dfrac{2}{θ}\) juu ya muda\(π≤θ≤2π\)

38) [T]\(r=\sin^2\left(\frac{θ}{2}\right)\) juu ya muda\(0≤θ≤π\)

Jibu: \(2\)vitengo

39) [T]\(r=2θ^2\) juu ya muda\(0≤θ≤π\)

40) [T]\(r=\sin(3\cos θ)\) juu ya muda\(0≤θ≤π\)

Jibu: \(4.39\)vitengo

Katika mazoezi 41 - 43, tumia fomu inayojulikana kutoka jiometri ili kupata eneo la kanda iliyoelezwa na kisha kuthibitisha kwa kutumia muhimu ya uhakika.

41)\(r=3\sin θ\) kwa muda\(0≤θ≤π\)

42)\(r=\sin θ+\cos θ\) kwa muda\(0≤θ≤π\)

Jibu: \(A=π\left(\frac{\sqrt{2}}{2}\right)^2=\dfrac{π}{2}\text{ units}^2\)na\(\displaystyle\frac{1}{2}∫^π_0(1+2\sin θ\cos θ)\,dθ=\frac{π}{2}\text{ units}^2\)

43)\(r=6\sin θ+8\cos θ\) kwa muda\(0≤θ≤π\)

Katika mazoezi 44 - 46, tumia formula inayojulikana kutoka jiometri ili kupata urefu wa curve na kisha kuthibitisha kutumia muhimu ya uhakika.

44)\(r=3\sin θ\) kwa muda\(0≤θ≤π\)

Jibu: \(C=2π\left(\frac{3}{2}\right)=3π\)vitengo na\(\displaystyle∫^π_03\,dθ=3π\) vitengo

45)\(r=\sin θ+\cos θ\) kwa muda\(0≤θ≤π\)

46)\(r=6\sin θ+8\cos θ\) kwa muda\(0≤θ≤π\)

Jibu: \(C=2π(5)=10π\)vitengo na\(\displaystyle∫^π_010\,dθ=10π\) vitengo

47) Thibitisha kwamba ikiwa\(y=r\sin θ=f(θ)\sin θ\) basi\(\dfrac{dy}{dθ}=f'(θ)\sin θ+f(θ)\cos θ.\)

Katika mazoezi 48 - 56, tafuta mteremko wa mstari wa tangent kwenye pembe ya polar\(r=f(θ)\). Hebu\(x=r\cos θ=f(θ)\cos θ\) na\(y=r\sin θ=f(θ)\sin θ\), hivyo equation polar sasa\(r=f(θ)\) imeandikwa katika fomu parametric.

48) Tumia ufafanuzi wa derivative\(\dfrac{dy}{dx}=\dfrac{dy/dθ}{dx/dθ}\) na utawala wa bidhaa ili kupata derivative ya equation polar.

Jibu: \(\dfrac{dy}{dx}=\dfrac{f′(θ)\sin θ+f(θ)\cos θ}{f′(θ)\cos θ−f(θ)\sin θ}\)

49)\(r=1−\sin θ; \; \left(\frac{1}{2},\frac{π}{6}\right)\)

50)\(r=4\cos θ; \; \left(2,\frac{π}{3}\right)\)

Jibu: Mteremko ni\(\frac{1}{\sqrt{3}}\).

51)\(r=8\sin θ; \; \left(4,\frac{5π}{6}\right)\)

52)\(r=4+\sin θ; \; \left(3,\frac{3π}{2}\right)\)

Jibu: Mteremko ni 0.

53)\(r=6+3\cos θ; \; (3,π)\)

54)\(r=4\cos(2θ);\) vidokezo vya majani

Jibu: Katika\((4,0),\) mteremko haujafafanuliwa. Katika\(\left(−4,\frac{π}{2}\right)\), mteremko ni 0.

55)\(r=2\sin(3θ);\) vidokezo vya majani

56)\(r=2θ; \; \left(\frac{π}{2},\frac{π}{4}\right)\)

Jibu: Mteremko haujafafanuliwa\(θ=\frac{π}{4}\).

57) Pata pointi\(−π≤θ≤π\) kwenye kipindi ambacho cardioid\(r=1−\cos θ\) ina mstari wa tangent wima au usawa.

58) Kwa cardioid\(r=1+\sin θ,\) kupata mteremko wa mstari wa tangent wakati\(θ=\frac{π}{3}\).

Jibu: Mteremko = -1.

Katika mazoezi 59 - 62, tafuta mteremko wa mstari wa tangent kwenye safu ya polar iliyotolewa kwa hatua iliyotolewa na thamani ya\(θ\).

59)\(r=3\cos θ,\; θ=\frac{π}{3}\)

60)\(r=θ, \; θ=\frac{π}{2}\)

Jibu: Mteremko ni\(\frac{−2}{π}\).

61)\(r=\ln θ, \; θ=e\)

62) [T] Tumia teknolojia:\(r=2+4\cos θ\) katika\(θ=\frac{π}{6}\)

Jibu: Jibu la Kikokotoo: -0.836.

Katika mazoezi 63 - 66, tafuta pointi ambazo curves zifuatazo za polar zina mstari wa usawa au wima wa tangent.

63)\(r=4\cos θ\)

64)\(r^2=4\cos(2θ)\)

Jibu: Horizontal tangent katika\(\left(±\sqrt{2},\frac{π}{6}\right), \; \left(±\sqrt{2},−\frac{π}{6}\right)\).

65)\(r=2\sin(2θ)\)

66) Cardioid\(r=1+\sin θ\)

Jibu: Horizontal tangents katika tangents\(\frac{π}{2},\, \frac{7π}{6},\, \frac{11π}{6}.\)
wima katika\(\frac{π}{6},\, \frac{5π}{6}\) na pia katika pole\((0,0)\).

67) Onyesha kwamba Curve\(r=\sin θ\tan θ\) (inayoitwa cissoid ya Diocles) ina mstari\(x=1\) kama asymptote wima.