16.2E: Mazoezi ya Sehemu ya 16.2

Last updated
Save as PDF

Page ID: 178927

Edwin “Jed” Herman & Gilbert Strang
OpenStax

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

1. Kweli au Uongo? Line muhimu$\displaystyle\int _C f(x,y)\,ds$ ni sawa na muhimu uhakika kama$C$ ni Curve laini defined juu$[a,b]$ na kama kazi$f$ ni kuendelea katika baadhi ya mkoa ambayo ina Curve$C$.

Jibu: Kweli

2. Kweli au Uongo? Vector kazi$\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}, \quad 0≤t≤1,$ na$\vecs r_2=(1−t)\,\hat{\mathbf i}+(1−t)^2\,\hat{\mathbf j}, \quad 0≤t≤1$, kufafanua moja oriented Curve.

3. Kweli au Uongo? $\displaystyle\int _{−C}(P\,dx+Q\,dy)=\int _C(P\,dx−Q\,dy)$

Jibu: Uongo

4. Kweli au Uongo? piecewise laini Curve$C$ lina idadi ya mwisho ya curves laini kwamba ni alijiunga pamoja mwisho hadi mwisho.

5. Kweli au Uongo? Ikiwa$C$ imetolewa na$x(t)=t,\quad y(t)=t, \quad 0≤t≤1$, basi$\displaystyle\int _Cxy\,ds=\int ^1_0t^2\,dt.$

Jibu: Uongo

Kwa mazoezi yafuatayo, tumia mfumo wa algebra ya kompyuta (CAS) kutathmini mstari wa mstari juu ya njia iliyoonyeshwa.

6. [T]$\displaystyle\int _C(x+y)\,ds$

$C:x=t,y=(1−t),z=0$kutoka$(0, 1, 0)$ kwa$(1, 0, 0)$

7. [T]$\displaystyle \int _C(x−y)ds$

$C:\vecs r(t)=4t\,\hat{\mathbf i}+3t\,\hat{\mathbf j}$lini$0≤t≤2$

Jibu: $\displaystyle\int _C(x−y)\,ds=10$

8. [T]$\displaystyle\int _C(x^2+y^2+z^2)\,ds$

$C:\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j}+8t\,\hat{\mathbf k}$lini$0≤t≤\dfrac{π}{2}$

9. [T] Tathmini$\displaystyle\int _Cxy^4\,ds$, ambapo$C$ ni nusu sahihi ya mduara$x^2+y^2=16$ na ni kupita katika mwelekeo clockwise.

Jibu: $\displaystyle\int _Cxy^4\,ds=\frac{8192}{5}$

10. [T] Kutathmini$\displaystyle\int _C4x^3ds$, ambapo C ni line sehemu kutoka$(−2,−1)$ kwa$(1, 2)$.

Kwa mazoezi yafuatayo, pata kazi iliyofanyika.

11. Kupata kazi iliyofanywa na uwanja vector$\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}$ juu ya chembe kusonga pamoja sehemu line kwamba huenda kutoka$(1,4,2)$ kwa$(0,5,1)$.

Jibu: $W=8$vitengo vya kazi

12. Kupata kazi kufanyika kwa mtu uzito 150 lb kutembea hasa mapinduzi moja juu ya mviringo, ond staircase ya Radius 3 ft kama mtu kuongezeka 10 ft.

13. Kupata kazi kufanyika kwa shamba nguvu$\vecs F(x,y,z)=−\dfrac{1}{2}x\,\hat{\mathbf i}−\dfrac{1}{2}y\,\hat{\mathbf j}+\dfrac{1}{4}\,\hat{\mathbf k}$ juu ya chembe kama hatua pamoja helix$\vecs r(t)=\cos t\,\hat{\mathbf i}+\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k}$ kutoka hatua$(1,0,0)$ kwa uhakika$(−1,0,3π)$.

Jibu: $W=\frac{3π}{4}$vitengo vya kazi

14. Pata kazi iliyofanywa na uwanja wa vector$\vecs{F}(x,y)=y\,\hat{\mathbf i}+2x\,\hat{\mathbf j}$ katika kusonga kitu kando ya njia$C$, ambayo hujiunga$(1, 0)$ na pointi na$(0, 1)$.

15. Kupata kazi kufanyika kwa nguvu$\vecs{F}(x,y)=2y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}+(x+y)\,\hat{\mathbf k}$ katika kusonga kitu pamoja Curve$\vecs r(t)=\cos(t)\,\hat{\mathbf i}+\sin(t)\,\hat{\mathbf j}+16\,\hat{\mathbf k}$, ambapo$0≤t≤2π$.

Jibu: $W=π$vitengo vya kazi

16. Pata wingi wa waya kwa sura ya mduara wa radius 2 unaozingatia (3, 4) na wiani wa wingi wa mstari$ρ(x,y)=y^2$.

Kwa mazoezi yafuatayo, tathmini ya mstari wa mstari.

17. Tathmini$\displaystyle\int_C\vecs F·d\vecs{r}$$\vecs{F}(x,y)=−1\,\hat{\mathbf j}$, wapi, na$C$ ni sehemu ya grafu ya$y=\frac{1}{2}x^3−x$ kutoka$(2,2)$ kwa$(−2,−2)$.

Jibu: $\displaystyle\int _C\vecs F·d\vecs{r}=4$vitengo vya kazi

18. Tathmini$\displaystyle\int _γ(x^2+y^2+z^2)^{−1}ds$,$γ$ wapi helix$x=\cos t,y=\sin t,z=t,$ na$0≤t≤T.$

19. Tathmini$\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz$ juu ya sehemu ya mstari kutoka$(1,1,1) $ kwa$(3,2,0).$

Jibu: $\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz=−1$

20. Hebu$C$ kuwa sehemu ya mstari kutoka hatua (0, 1, 1) hadi kumweka (2, 2, 3). Tathmini ya mstari muhimu$\displaystyle\int _Cy\,ds.$

21. [T] Tumia mfumo wa kompyuta algebra kutathmini mstari muhimu$\displaystyle\int _Cy^2\,dx+x\,dy$, wapi$C$ safu ya parabola$x=4−y^2$ kutoka$(−5, −3)$ kwa$(0, 2)$.

Jibu: $\displaystyle\int _C(y^2)\,dx+(x)\,dy=\dfrac{245}{6}$

22. [T] Tumia mfumo wa algebra ya kompyuta kutathmini mstari muhimu$\displaystyle\int _C (x+3y^2)\,dy$ juu ya njia$C$ iliyotolewa na$x=2t,y=10t,$ wapi$0≤t≤1.$

23. [T] Matumizi CAS kutathmini line muhimu$\displaystyle\int _C xy\,dx+y\,dy$ juu ya njia$C$ iliyotolewa na$x=2t,y=10t$, ambapo$0≤t≤1$.

Jibu: $\displaystyle\int _Cxy\,dx+y\,dy=\dfrac{190}{3}$

24. Tathmini mstari muhimu$\displaystyle\int _C(2x−y)\,dx+(x+3y)\,dy$, ambapo$C$ uongo kando$x$ -axis kutoka$x=0$ kwa$x=5$.

26. [T] Tumia CAS kutathmini$\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds$, ambapo$C$ hufafanuliwa na equations parametric$x=t,y=t$, kwa$1≤t≤5.$

Jibu: $\displaystyle\int _C\frac{y}{2x^2−y^2}\,ds=\sqrt{2}\ln 5$

27. [T] Tumia CAS kutathmini$\displaystyle\int _Cxy\,ds$, ambapo$C$ hufafanuliwa na equations parametric$x=t^2,y=4t$, kwa$0≤t≤1.$

Katika mazoezi yafuatayo, tafuta kazi iliyofanywa na shamba la nguvu$\vecs F$ kwenye kitu kinachohamia njia iliyoonyeshwa.

28. $\vecs{F}(x,y)=−x \,\hat{\mathbf i}−2y\,\hat{\mathbf j}$

$C:y=x^3$kutoka$(0, 0)$ kwa$(2, 8)$

Jibu: $W=−66$vitengo vya kazi

29. $\vecs{F}(x,y)=2x\,\hat{\mathbf i}+y\,\hat{\mathbf j}$

$C$<:kinyume chake karibu na pembetatu na vertices$(0, 0), (1, 0), $ na$(1, 1)$

30. $\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}−5z\,\hat{\mathbf k}$

$C:\vecs r(t)=2\cos t\,\hat{\mathbf i}+2\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k},\; 0≤t≤2π$

Jibu: $W=−10π^2$vitengo vya kazi

31. Hebu$\vecs F$ kuwa vector shamba$\vecs{F}(x,y)=(y^2+2xe^y+1)\,\hat{\mathbf i}+(2xy+x^2e^y+2y)\,\hat{\mathbf j}$. Compute kazi ya muhimu$\displaystyle\int _C\vecs F·d\vecs{r}$,$C$ wapi njia$\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j},\quad 0≤t≤\dfrac{π}{2}$.

32. Compute kazi iliyofanywa kwa nguvu$\vecs F(x,y,z)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}−z\,\hat{\mathbf k}$ njiani$\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+t^3\,\hat{\mathbf k}$, ambapo$0≤t≤1$.

Jibu: $W=2$vitengo vya kazi

33. Tathmini$\displaystyle\int _C\vecs F·d\vecs{r}$, wapi$\vecs{F}(x,y)=\dfrac{1}{x+y}\,\hat{\mathbf i}+\dfrac{1}{x+y}\,\hat{\mathbf j}$ na$C$ ni sehemu ya mduara wa kitengo kinachoenda kinyume chake kutoka$(1,0)$ kwa$(0, 1)$.

34. Nguvu$\vecs F(x,y,z)=zy\,\hat{\mathbf i}+x\,\hat{\mathbf j}+z^2x\,\hat{\mathbf k}$ vitendo juu ya chembe kwamba safari kutoka asili kwa uhakika$(1, 2, 3)$. Tumia kazi iliyofanyika ikiwa chembe inasafiri:

kando ya njia$(0,0,0)→(1,0,0)→(1,2,0)→(1,2,3)$ pamoja na makundi ya mstari wa moja kwa moja kujiunga na kila jozi ya mwisho;
pamoja na mstari wa moja kwa moja kujiunga na pointi za awali na za mwisho.
Je, kazi hiyo ni sawa katika njia mbili?
$clipboard_e7c787ff46860b19ec57c0669a08914af.png$

Jibu: a.$W=11$ vitengo vya kazi;
b.$W=\dfrac{39}{4}=9\frac{3}{4}$ vitengo vya kazi;
c. hakuna

35. Kupata kazi iliyofanywa na uwanja vector$\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}$ juu ya chembe kusonga pamoja sehemu line kwamba huenda kutoka$(1, 4, 2)$ kwa$(0, 5, 1).$

36. Ni kazi gani inahitajika kuhamisha kitu katika uwanja wa vector$\vecs{F}(x,y)=y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}$ kando ya sehemu ya juu ya ellipse$\dfrac{x^2}{4}+y^2=1$ kutoka$(2, 0)$ kwa$(−2,0)$?

Jibu: $W=2π$vitengo vya kazi

37. Shamba la vector linatolewa na$\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}$. Tathmini ya mstari muhimu wa shamba karibu na mduara wa kitengo cha radius kilichopita kwa mtindo wa saa.

38. Tathmini mstari muhimu ya kazi scalar$xy$ pamoja njia parabolic$y=x^2$ kuunganisha asili kwa uhakika$(1, 1)$.

Jibu: $\displaystyle\int _C f\,ds=\dfrac{25\sqrt{5}+1}{120}$

39. Kupata$\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy$ pamoja$C: y=3x$ kutoka$(0, 0)$ kwa$(1, 3).$

40. Kupata$\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy$ pamoja$C: y^2=9x$ kutoka$(0, 0)$ kwa$(1, 3).$

Jibu: $\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy=6.15$

Kwa mazoezi yafuatayo, tumia CAS ili kutathmini mchanganyiko wa mstari uliotolewa.

41. [T] Tathmini$\vecs F(x,y,z)=x^2z\,\hat{\mathbf i}+6y\,\hat{\mathbf j}+yz^2\,\hat{\mathbf k}$, ambapo$C$ inawakilishwa na$\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+\ln t \,\hat{\mathbf k},1≤t≤3$.

42. [T] Kutathmini line muhimu$\displaystyle\int _γxe^y\,ds$ ambapo,$γ$ ni safu ya Curve$x=e^y$ kutoka$(1,0)$ kwa$(e,1)$.

Jibu: $\displaystyle\int _γxe^y\,ds≈7.157$

43. [T] Tathmini muhimu$\displaystyle\int _γxy^2\,ds$, wapi$γ$ pembetatu na vipeo$(0, 1, 2), (1, 0, 3)$, na$(0,−1,0)$.

44. [T] Kutathmini line muhimu$\displaystyle\int _γ(y^2−xy)\,dx$, ambapo$γ$ ni Curve$y=\ln x$ kutoka$(1, 0)$ kuelekea$(e,1)$.

Jibu: $\displaystyle\int _γ(y^2−xy)\,dx≈−1.379$

45. [T] Kutathmini line muhimu$\displaystyle\int_γ xy^4\,ds$, ambapo$γ$ ni nusu ya haki ya mduara$x^2+y^2=16$.

46. [T] Tathmini$\int C \vecs F⋅d\vecs{r},\int C \vecs F·d\vecs{r},$ wapi$\vecs F(x,y,z)=x^2y\,\mathbf{\hat i}+(x−z)\,\mathbf{\hat j}+xyz\,\mathbf{\hat k}$ na

$C: \vecs r(t)=t\,\mathbf{\hat i}+t^2\,\mathbf{\hat j}+2\,\mathbf{\hat k},0≤t≤1$.

Jibu: $\displaystyle\int _C \vecs F⋅d\vecs{r}≈−1.133$vitengo vya kazi

47. Tathmini$\displaystyle\int _C \vecs F⋅d\vecs{r}$, wapi$\vecs{F}(x,y)=2x\sin y\,\mathbf{\hat i}+(x^2\cos y−3y^2)\,\mathbf{\hat j}$ na

$C$ni njia yoyote kutoka$(−1,0)$ kwa$(5, 1)$.

48. Kupata line muhimu ya$\vecs F(x,y,z)=12x^2\,\mathbf{\hat i}−5xy\,\mathbf{\hat j}+xz\,\mathbf{\hat k}$ juu ya njia$C$ inavyoelezwa na$y=x^2, z=x^3$ kutoka hatua$(0, 0, 0)$ hadi hatua$(2, 4, 8)$.

Jibu: $\displaystyle\int _C \vecs F⋅d\vecs{r}≈22.857$vitengo vya kazi

49. Kupata line muhimu ya$\displaystyle\int _C(1+x^2y)\,ds$, wapi$C$ duaradufu$\vecs r(t)=2\cos t\,\mathbf{\hat i}+3\sin t\,\mathbf{\hat j}$ kutoka$0≤t≤π.$

Kwa mazoezi yafuatayo, tafuta mtiririko.

50. Compute flux ya$\vecs{F}=x^2\,\mathbf{\hat i}+y\,\mathbf{\hat j}$ katika sehemu ya mstari kutoka$(0, 0)$ kwa$(1, 2).$

Jibu: $\text{flux}=−\frac{1}{3}$

51. Hebu$\vecs{F}=5\,\mathbf{\hat i}$ na hebu$C$ kuwa Curve$y=0,$ na$0≤x≤4$. Pata flux kote$C$.

52. Hebu$\vecs{F}=5\,\mathbf{\hat j}$ na hebu$C$ kuwa Curve$y=0,$ na$0≤x≤4$. Pata flux kote$C$.

Jibu: $\text{flux}=-20$

53. Hebu$\vecs{F}=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j}$ na uache$C: \vecs r(t)=\cos t\,\mathbf{\hat i}+\sin t\,\mathbf{\hat j}$$0≤t≤2π$. Tumia flux kote$C$.

54. Hebu$\vecs{F}=(x^2+y^3)\,\mathbf{\hat i}+(2xy)\,\mathbf{\hat j}$. Tumia flux$\vecs F$ iliyoelekezwa kinyume chake kwenye pembe$C: x^2+y^2=9.$

Jibu: $\text{flux}=0$

Jaza mazoezi yote kama ilivyoelezwa.

55. Kupata line muhimu ya$\displaystyle\int _C z^2\,dx+y\,dy+2y\,dz,$ ambapo$C$ lina sehemu mbili:$C_1$ na$C_2.$$C_1$ ni makutano ya silinda$x^2+y^2=16$ na ndege$z=3$ kutoka$(0, 4, 3)$ kwa$(−4,0,3).$$C_2$ ni line sehemu kutoka$(−4,0,3)$ kwa$(0, 1, 5)$.

56. Spring ni ya waya nyembamba inaendelea katika sura ya helix mviringo$x=2\cos t,\;y=2\sin t,\;z=t.$ Kupata wingi wa zamu mbili za spring ikiwa waya ina wiani wa mara kwa mara wa$ρ$ gramu kwa cm.

Jibu: $m=4πρ\sqrt{5}$gramu

57. Waya mwembamba hupigwa katika sura ya semicircle ya radius$a$. Ikiwa wiani wa molekuli wa mstari kwenye hatua$P$ ni sawa sawa na umbali wake kutoka kwenye mstari kupitia mwisho, pata wingi wa waya.

58. kitu hatua katika uwanja nguvu$\vecs F(x,y,z)=y^2\,\mathbf{\hat i}+2(x+1)y\,\mathbf{\hat j}$ kinyume chake kutoka hatua$(2, 0)$ pamoja njia elliptical$x^2+4y^2=4$ kwa$(−2,0)$, na kurudi kwa uhakika$(2, 0)$ pamoja$x$ -axis. Ni kazi gani inayofanywa na shamba la nguvu kwenye kitu?

Jibu: $W=0$vitengo vya kazi

59. Find kazi kufanyika wakati kitu hatua katika uwanja nguvu$\vecs F(x,y,z)=2x\,\mathbf{\hat i}−(x+z)\,\mathbf{\hat j}+(y−x)\,\mathbf{\hat k}$ kando ya njia iliyotolewa na$\vecs r(t)=t^2\,\mathbf{\hat i}+(t^2−t)\,\mathbf{\hat j}+3\,\mathbf{\hat k}, \; 0≤t≤1.$

60. Ikiwa uwanja wa nguvu wa inverse$\vecs F$ unatolewa na$\vecs F(x,y,z)=\dfrac{k}{‖r‖^3}r$, wapi$k$ mara kwa mara, pata kazi iliyofanywa na$\vecs F$ kama hatua yake ya maombi inakwenda kando ya$x$ -axis kutoka$A(1,0,0)$ kwa$B(2,0,0)$.

Jibu: $W=\frac{k}{2}$vitengo vya kazi

61. David na Sandra mpango wa kutathmini line muhimu$\displaystyle\int _C\vecs F·d\vecs{r}$ kando ya njia katika$xy$ -ndege kutoka$(0, 0)$ kwa$(1, 1)$. Shamba la nguvu ni$\vecs{F}(x,y)=(x+2y)\,\mathbf{\hat i}+(−x+y^2)\,\mathbf{\hat j}$. Daudi anachagua njia inayoendesha kando ya$x$ mhimili kutoka$(0, 0)$ kwenda$(1, 0)$ na kisha anaendesha kando ya mstari wa wima$x=1$ kutoka$(1, 0)$ hadi hatua ya mwisho$(1, 1)$. Sandra anachagua njia moja kwa moja kwenye mstari wa diagonal$y=x$ kutoka$(0, 0)$ kwa$(1, 1)$. Nani mstari muhimu ni mkubwa na kwa kiasi gani?

Wachangiaji

Template:contribopenstaxcalc