Calculating Volume Using Triple Integrals in Spherical Coordinates

[mhs11]

hi all

how can i find the volume of the solid that lies within the sphere x^2+y^2+z^2=36 , above the xy plane, and outside the cone z=7sqrt(x^2+y^2) .

your help is very much appreiated

 

[tiny-tim]

Welcome to PF!

Hi mhs1! Welcome to PF!

(try using the X² tag just above the Reply box )

Either split it into "vertical" cylinders of thickness dr, or split it into "horizontal" discs-with-holes-in of height dz.

 

[mhs11]

thnx 4 ur reply

but i didn't get it

how can if find the boundaries

 

[tiny-tim]

… how can if find the boundaries

Do you mean the limits of integration?

If you integrate over r (= √(x² + y²)), do it from 0 to the maximum value of r.

If you integrate over z, do it from 0 to the maximum value of z.

 

[mhs11]

i did the following:

0 ≤ σ ≤ 6
σ^2 dσ= σ^3 /3 = 72

0 ≤ q ≤ 2π

dq= q = 2π

arctan 7/√50 ≤ Φ ≤ π

sinΦ dΦ= -cosΦ= 1+cos(arctan 7/√50 )

then i multiply them

(1+cos(arctan 7/√50 ))*2π *72=773.8884482

but when i enter it it gives me that it is wrong

 

[tiny-tim]

i did the following:

0 ≤ σ ≤ 6
σ^2 dσ= σ^3 /3 = 72

0 ≤ q ≤ 2π

dq= q = 2π

arctan 7/√50 ≤ Φ ≤ π

sinΦ dΦ= -cosΦ= 1+cos(arctan 7/√50 )

then i multiply them

(1+cos(arctan 7/√50 ))*2π *72=773.8884482

but when i enter it it gives me that it is wrong

I'm not following this at all.

What is σ ?

What is σ²dσ supposed to be?

What is 7/√50 ?

What are you trying to integrate?

 

[mhs11]

i'm trying to find the volume using shperical coordinate

 

[tiny-tim]

i'm trying to find the volume using shperical coordinate

(I would have used either cylindrical or Cartesian coordinates.)

(and it's arctan7, not arctan 7/√50, though it is arcsin7/√50)

ok, write it out properly this time … what is the basic formula for volume, using spherical coordinates?

(oh, and have an integral: ∫ and a theta: θ and a phi: φ )