Triple integral-spherical coordinates

[mcc123pa]

Homework Statement

Here is exactly what the problem says:
Let T be a solid of density one that lies below x^2+y^2+z^2=9 and above the XY-plane.
a) Use the triple integral to find its mass.
b) Find the centroid.

Homework Equations

I believe that this should be an integral in spherical coordinates. I know the relevant equation is p^2sin(phi) dpd(phi)d(theta).

The Attempt at a Solution

Obviously the first step is to set up the triple integral in spherical coordinates. I set x^2+y^2+z^2=9 and changed it to p^2=9. This left me with the limits of integration for p being from 0 to 3. I know that theta's integral is normally from 0 to 2(pie). I have no clue how to solve for the 'phi' limits and obviously I can't complete the problem until I do. Please advise as soon as possible what I should do. Thanks!

 

[tiny-tim]

welcome to pf!

hi mcc123pa! welcome to pf!

(have a theta: θ and a phi: φ and a pi: π and try using the X² icon just above the Reply box )

φ is the latitude (or co-latitude), and it goes from the pole to the equator, so that's … ?

 

[Matterwave]

This is not the usual definition of φ in spherical coordinates. Usually φ is the polar coordinate (analogous to longitude, but just starting at 0 and ending at 360, instead of being split between west and east), and θ is the co-latitude (90 degrees-latitude, with latitudes being negative below the x-y plane).

 

[tiny-tim]

Hi Matterwave!

No, it's ok …

engineers do it the other way round! ​

 

[mcc123pa]

Hi everyone- thanks for the help so far...

tiny-tim,

From the pole to the equator, that's pie/2. Is that correct?

So I guess in the case that is correct, the limits are as follows:

p = from 0 to 3

phi = from 0 to (pie)/2

theta = from 0 to (2pie)

Please confirm if I have this right everyone. Thanks for your help!

 

[LCKurtz]

Hi everyone- thanks for the help so far...

tiny-tim,

From the pole to the equator, that's pie/2. Is that correct?

So I guess in the case that is correct, the limits are as follows:

p = from 0 to 3

phi = from 0 to (pie)/2

theta = from 0 to (2pie)

Please confirm if I have this right everyone. Thanks for your help!

That's right. But it is "pi", not "pie".

 

[mcc123pa]

Thanks LCKurtz! I'll remember it's pi not pie from now on...thanks for the tip!