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

Question

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 · Answer

@ganeshie8

ganeshie8 · Answer

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

anonymous · Answer

as my limits for triple integration?

ganeshie8 · Answer

Yes

anonymous · Answer

p is dx θ is dy ϕ is dz?

anonymous · Answer

whats the functions im integrating?

ganeshie8 · Answer

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

ganeshie8 · Answer

@jyar  what do you know about spherical coordinates ?

anonymous · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

angles projected from the xyz plane

loser66 · Answer

.

anonymous · Answer

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

ganeshie8 · Answer

|dw:1436648757792:dw|

ganeshie8 · Answer

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

anonymous · Answer

oh okay i understand that

ganeshie8 · Answer

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

ganeshie8 · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

is θ always 0 to 2pi?

ganeshie8 · Answer

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

anonymous · Answer

2 full revolution would be 4pi?

ganeshie8 · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

yeah i see that

ganeshie8 · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

pi/3 to 0?

ganeshie8 · Answer

z axis is perpendicular to xy plane, yes ?

anonymous · Answer

yes

ganeshie8 · Answer

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

anonymous · Answer

its looks 60 degrees

anonymous · Answer

its not that ?

ganeshie8 · Answer

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

ganeshie8 · Answer

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