Triple integral-spherical coordinates

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In summary, the problem asks for the mass of a solid that lies below x^2+y^2+z^2=9 and above the XY-plane. It also asks for the centroid of the solid.
  • #1
mcc123pa
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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!
 
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  • #2
welcome to pf!

hi mcc123pa! welcome to pf! :smile:

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

φ is the latitude (or co-latitude), and it goes from the pole to the equator, so that's … ? :smile:
 
  • #3


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).
 
  • #4
Hi Matterwave! :smile:

No, it's ok …

engineers do it the other way round! :biggrin:​
 
  • #5
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!
 
  • #6
mcc123pa said:
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".
 
  • #7
Thanks LCKurtz! I'll remember it's pi not pie from now on...thanks for the tip!
 

1. What is a triple integral in spherical coordinates?

A triple integral in spherical coordinates is a mathematical concept used to calculate the volume of a three-dimensional region in space. It involves integrating a function over a spherical coordinate system, which uses angles and a radial distance from a fixed point to specify a point in space.

2. How do you set up a triple integral in spherical coordinates?

To set up a triple integral in spherical coordinates, you need to determine the limits of integration for each of the three variables: ρ, θ, and φ. ρ represents the distance from the origin to a point, θ represents the angle in the xy-plane, and φ represents the angle from the positive z-axis. The integral is then written as ∫∫∫ f(ρ, θ, φ) ρ²sinφ dρ dθ dφ, where f(ρ, θ, φ) is the function being integrated.

3. What are the advantages of using spherical coordinates for triple integrals?

Spherical coordinates are advantageous for triple integrals because they allow for integration over a variety of shapes, including spheres, cones, and cylinders. They also simplify the calculation of integrals involving functions with radial symmetry, as the limits of integration for ρ are constant.

4. How do you convert a triple integral from Cartesian coordinates to spherical coordinates?

To convert a triple integral from Cartesian coordinates to spherical coordinates, you can use the following formulas:

ρ = √(x² + y² + z²)

θ = arctan(y/x)

φ = arccos(z/ρ)

Substituting these values into the original integral and replacing dxdydz with ρ²sinφ dρ dθ dφ will give you the equivalent integral in spherical coordinates.

5. How do you evaluate a triple integral in spherical coordinates?

Evaluating a triple integral in spherical coordinates involves first setting up the integral with the appropriate limits of integration and then solving it using basic integration techniques. It is important to carefully consider the limits and make any necessary substitutions to simplify the integral before integrating. Additionally, using a graphing calculator or computer software can help with evaluating the integral and visualizing the three-dimensional region being integrated.

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