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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
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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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Try \(\rho : ~0\to 1 \) \(\theta : ~0\to 2\pi \) \(\phi : ~\frac{\pi}{3}\to \frac{\pi}{2}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0as my limits for triple integration?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0p is dx θ is dy ϕ is dz?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0whats the functions im integrating?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Hmm @Loser66 I didn't get your question..

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3@jyar what do you know about spherical coordinates ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0there related to the cartesian coordinates and its in 3d points (r, θ , ϕ )

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3what do \(\rho, \theta\) and \(\phi\) represent ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0angles projected from the xyz plane

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3dw:1436648757792:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3For 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh okay i understand that

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3For our specific problem, it is easy to see that \(\rho\) varies from 0 to 1 because the radius of sphere is 1

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Also \(\theta\) varies from 0 to 2pi is also trivial

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3you need to do some work to figure out the bounds for \(\phi\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is θ always 0 to 2pi?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3It 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.02 full revolution would be 4pi?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3You never want to do 2 full revolutions as that might duplicate the volume

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3btw you're correct about 4pi being two full revolutions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh okay so when finding the limits of ϕ what would i do

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3the part of sphere between z=1/2 and z=0 looks like below? dw:1436649362434:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3z=0 represents the xy plane, whats the angle \(\phi\) for xy plane ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Remember, \(\phi\) is the angle from z axis

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3z axis is perpendicular to xy plane, yes ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3so whats the angle between xy plane and positive z axis ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0its looks 60 degrees

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Easy, xy plane makes 90 degrees with the positive z axis.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3dw:1436649826870:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3that means pi/2 is the ending angle lets find the starting angle

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0its not under pi.2 its a little further away...so 17pi/12?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3dw:1436649896094:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3see if you can find \(\theta\) in that right triangle

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Honestly I have no idea how you got 17pi/12

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Yes starting angle is 60 degrees, which is same as pi/3

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3so do the bounds make sense ? \(\rho : ~0\to 1 \) \(\theta : ~0\to 2\pi \) \(\phi : ~\frac{\pi}{3}\to \frac{\pi}{2}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3The volume is given by \[\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \int\limits_{\pi/3}^{\pi/2}~1~\color{blue}{\rho^2\sin\phi~d\phi~d\theta~d\rho}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok! the first integrations foes with p^2dp, second sinϕ dϕ, third dθ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3It doesn't matter, only the bounds and the differentials need to agree

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Now that we have moved to spherical, \(\large \color{blue}{\rho^2 \sin\phi}\) is your integrand here

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok i got that...could we do the first integration

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3start by working the inner integral : \[\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the 1 intregrated becomes just ϕ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3integrate below and plug it in \[\large \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3thats same as \[\large \color{red}{\int\limits_{\pi/3}^{\pi/2} \rho^2\sin\phi~d\phi}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3you can pull out \(\color{red}{\rho^2}\) because it is constant with respect to \(\phi\) : \[\large \color{red}{\rho^2\int\limits_{\pi/3}^{\pi/2} \sin\phi~d\phi}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i got (1/162) without pulling the p^2 out

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0pulling the p^2 i get (1/2)p^2

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3\[\large \color{red}{\rho^2\int\limits_{\pi/3}^{\pi/2} \sin\phi~d\phi} = \color{red}{\rho^2 (\cos\phi)~ \Bigg_{\pi/3}^{\pi/2} } = \color{red}{\rho^2 [(0\frac{1}{2})]} = \color{red}{\frac{1}{2}\rho^2}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok yes...for the next integration i \[\int\limits_{0}^{2\pi} (1/2)p^2 dp\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3plugging that in the volume integral we get \[\large V = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{ \frac{1}{2}\rho^2}~d\theta~d\rho\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Nope, next we work : \[\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3keepin mind \(0\to 2\pi\) are bounds for \(\theta\), not \(\rho\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh okay so those bounds also how the order of work.... for the next intregation i got (4pi/3)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3\[\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta} = ?\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i pulled out the (1/2) and the inside i got (2pi^3/3) = (8pi/3)= (1/2)(8pi/3)?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3looks wrong, you can pull out entire thing, everythng is constant there

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3\[\large \int\limits_{0}^{2\pi} (1/2)p^2 d\color{red}{\theta} = (1/2)p^2\int\limits_{0}^{2\pi} 1 d\color{red}{\theta} = ? \]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Yes, plug that in the volume integral

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what if i didnt pull everything out as a constant and instead integrated would that be all wrong?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3plugging that in the volume integral we get \[\begin{align}\large V &= \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{\int\limits_{\pi/3}^{\pi/2}~1~\rho^2\sin\phi~d\phi}~d\theta~d\rho = \int\limits_{0}^1 ~\int\limits_{0}^{2\pi}~ \color{red}{ \frac{1}{2}\rho^2}~d\theta~d\rho\\~\\ &=\int\limits_{0}^1 ~\pi \rho^2 ~d\rho\\~\\ &= ? \end{align}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0regarding my question for the second integration if i didnt pull everything out as a constant and intregrated evrthhing instead would that be wrong?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3can you show me how exactly are you "integrating" everything ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{0}^{2\pi}(1/2)p^2 d\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait i think i got it ! wow thank you soo much! so helpful

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3feel free to ask if you have any questions :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0will you be online for rest of today?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3il be around for 1 hour or so, feel free to tag me in ur questions
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