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## anonymous one year ago MEDAL***Please help use spherical coordinates to find the volume cut out from the sphere x^2+y^2+z^2=1 by the planes z=1/2 and z=0

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1. anonymous

@ganeshie8

2. ganeshie8

Try $$\rho : ~0\to 1$$ $$\theta : ~0\to 2\pi$$ $$\phi : ~\frac{\pi}{3}\to \frac{\pi}{2}$$

3. anonymous

as my limits for triple integration?

4. ganeshie8

Yes

5. anonymous

p is dx θ is dy ϕ is dz?

6. anonymous

whats the functions im integrating?

7. ganeshie8

Hmm @Loser66 I didn't get your question..

8. ganeshie8

@jyar what do you know about spherical coordinates ?

9. anonymous

there related to the cartesian coordinates and its in 3d points (r, θ , ϕ )

10. ganeshie8

what do $$\rho, \theta$$ and $$\phi$$ represent ?

11. anonymous

angles projected from the xyz plane

12. Loser66

.

13. anonymous

p^2=x^2+y^2+z^2

14. ganeshie8

|dw:1436648757792:dw|

15. ganeshie8

For any point $$(\rho, \theta, \phi)$$ in space, $$\rho$$ is the "distance" from origin $$\theta$$ is the angle in xy plane with the positive x axis $$\phi$$ is the angle with positive z axis

16. anonymous

oh okay i understand that

17. ganeshie8

For our specific problem, it is easy to see that $$\rho$$ varies from 0 to 1 because the radius of sphere is 1

18. ganeshie8

Also $$\theta$$ varies from 0 to 2pi is also trivial

19. ganeshie8

you need to do some work to figure out the bounds for $$\phi$$

20. anonymous

is θ always 0 to 2pi?

21. ganeshie8

It depends, from the diagram you can see that $$\theta$$ is the angle in xy plane. $$\theta ~: ~0\to 2\pi$$ means this angle is swept one full revolution

22. anonymous

2 full revolution would be 4pi?

23. ganeshie8

You never want to do 2 full revolutions as that might duplicate the volume

24. ganeshie8

btw you're correct about 4pi being two full revolutions

25. anonymous

oh okay so when finding the limits of ϕ what would i do

26. ganeshie8

the part of sphere between z=1/2 and z=0 looks like below? |dw:1436649362434:dw|

27. anonymous

yeah i see that

28. ganeshie8

z=0 represents the xy plane, whats the angle $$\phi$$ for xy plane ?

29. ganeshie8

Remember, $$\phi$$ is the angle from z axis

30. anonymous

pi/3 to 0?

31. ganeshie8

z axis is perpendicular to xy plane, yes ?

32. anonymous

yes

33. ganeshie8

so whats the angle between xy plane and positive z axis ?

34. anonymous

its looks 60 degrees

35. anonymous

its not that ?

36. ganeshie8

Easy, xy plane makes 90 degrees with the positive z axis.

37. ganeshie8

|dw:1436649826870:dw|

38. ganeshie8

that means pi/2 is the ending angle lets find the starting angle

39. anonymous

its not under pi.2 its a little further away...so 17pi/12?

40. ganeshie8

|dw:1436649896094:dw|

41. ganeshie8

see if you can find $$\theta$$ in that right triangle

42. ganeshie8

Honestly I have no idea how you got 17pi/12

43. anonymous

60 i got

44. ganeshie8

Yes starting angle is 60 degrees, which is same as pi/3

45. anonymous

yes

46. ganeshie8

so do the bounds make sense ? $$\rho : ~0\to 1$$ $$\theta : ~0\to 2\pi$$ $$\phi : ~\frac{\pi}{3}\to \frac{\pi}{2}$$

47. ganeshie8

The volume is given by $\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \int\limits_{\pi/3}^{\pi/2}~1~\color{blue}{\rho^2\sin\phi~d\phi~d\theta~d\rho}$

48. anonymous

ok! the first integrations foes with p^2dp, second sinϕ dϕ, third dθ?

49. ganeshie8

It doesn't matter, only the bounds and the differentials need to agree

50. ganeshie8

Now that we have moved to spherical, $$\large \color{blue}{\rho^2 \sin\phi}$$ is your integrand here

51. anonymous

ok i got that...could we do the first integration

52. ganeshie8

start by working the inner integral : $\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho$

53. ganeshie8

that red part

54. anonymous

the 1 intregrated becomes just ϕ?

55. ganeshie8

integrate below and plug it in $\large \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}$

56. ganeshie8

thats same as $\large \color{red}{\int\limits_{\pi/3}^{\pi/2} \rho^2\sin\phi~d\phi}$

57. ganeshie8

you can pull out $$\color{red}{\rho^2}$$ because it is constant with respect to $$\phi$$ : $\large \color{red}{\rho^2\int\limits_{\pi/3}^{\pi/2} \sin\phi~d\phi}$

58. anonymous

i got (-1/162) without pulling the p^2 out

59. anonymous

pulling the p^2 i get (1/2)p^2

60. anonymous

(-1/2)

61. ganeshie8

$\large \color{red}{\rho^2\int\limits_{\pi/3}^{\pi/2} \sin\phi~d\phi} = \color{red}{\rho^2 (-\cos\phi)~ \Bigg|_{\pi/3}^{\pi/2} } = \color{red}{\rho^2 [-(0-\frac{1}{2})]} = \color{red}{\frac{1}{2}\rho^2}$

62. anonymous

ok yes...for the next integration i $\int\limits_{0}^{2\pi} (1/2)p^2 dp$

63. ganeshie8

plugging that in the volume integral we get $\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{ \frac{1}{2}\rho^2}~d\theta~d\rho$

64. ganeshie8

Nope, next we work : $\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta}$

65. ganeshie8

keepin mind $$0\to 2\pi$$ are bounds for $$\theta$$, not $$\rho$$

66. anonymous

oh okay so those bounds also how the order of work.... for the next intregation i got (4pi/3)

67. ganeshie8

$\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta} = ?$

68. anonymous

i pulled out the (1/2) and the inside i got (2pi^3/3) = (8pi/3)= (1/2)(8pi/3)?

69. ganeshie8

looks wrong, you can pull out entire thing, everythng is constant there

70. ganeshie8

$\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta} = (1/2)p^2\int\limits_{0}^{2\pi} 1 d\color{red}{\theta} = ?$

71. anonymous

(pi) p^2?

72. ganeshie8

Yes, plug that in the volume integral

73. anonymous

what if i didnt pull everything out as a constant and instead integrated would that be all wrong?

74. ganeshie8

plugging that in the volume integral we get \begin{align}\large V &= \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{ \frac{1}{2}\rho^2}~d\theta~d\rho\\~\\ &=\int\limits_{0}^1 ~\pi \rho^2 ~d\rho\\~\\ &= ? \end{align}

75. anonymous

pi/3?

76. ganeshie8

Yep!

77. anonymous

regarding my question for the second integration if i didnt pull everything out as a constant and intregrated evrthhing instead would that be wrong?

78. ganeshie8

can you show me how exactly are you "integrating" everything ?

79. anonymous

$\int\limits_{0}^{2\pi}(1/2)p^2 d$

80. anonymous

wait i think i got it ! wow thank you soo much! so helpful

81. ganeshie8

d what ?

82. anonymous

d theta

83. ganeshie8

feel free to ask if you have any questions :)

84. anonymous

will you be online for rest of today?

85. ganeshie8

il be around for 1 hour or so, feel free to tag me in ur questions

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