Question on limits used in triple integral for volume of a sphere

In summary, the incorrect jacobian was used to derive the volume of the sphere. The correct jacobian was found using the limits for theta and phi angles, which are opposite of the convention used in spherical coordinates. I was able to find the correct formula using this information, but why do we use opposite limits for the two coordinates?
  • #1
redpen
5
0

Homework Statement



I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits.

Homework Equations



What I don't get is why we use 2pi to 0 for phi and pi to 0 for theta, when surely the definition of spherical coordinates uses the opposite limits for the two coordinates.

The Attempt at a Solution



Here http://en.wikipedia.org/wiki/Multip...atical_applications_-_Calculations_of_volume" the correct formula is derived using these limits, so I know it works. But why?


Any help appreciated!
 
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  • #2
redpen said:

Homework Statement



I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits.

Homework Equations



What I don't get is why we use 2pi to 0 for phi and pi to 0 for theta, when surely the definition of spherical coordinates uses the opposite limits for the two coordinates.

The Attempt at a Solution



Here http://en.wikipedia.org/wiki/Multip...atical_applications_-_Calculations_of_volume" the correct formula is derived using these limits, so I know it works. But why?


Any help appreciated!

It depends on your definition of the coordinates. From your above volume element, I presume you're using the convention that the theta angle is the "polar angle" i.e. the one measuring the angle from the z axis, and the phi angle is the "azimuthal angle" i.e. the angle measured in the x-y plane, measured from the x axis.

If these are your conventions, then the theta angle measures from +z to -z (i.e. between 0 and pi) whilst the phi angle covers the whole circle in the x-y plane (i.e. 0 to 2 pi)

Note that there are different conventions used; the convention I mention is used by most applied mathmaticians, engineers, physicists. However, pure mathematicians use the convention that the theta angle is the azimuthal angle, and the phi angle is the polar angle.
 
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  • #3
I got it! Yes, I hade got the jacobian wrong. Thanks for the help!
 

FAQ: Question on limits used in triple integral for volume of a sphere

What is a triple integral?

A triple integral is a type of integral that is used to find the volume of a three-dimensional solid within a given boundary. It involves integrating a function over three variables, typically representing the x, y, and z coordinates.

How is a triple integral used to find the volume of a sphere?

In order to find the volume of a sphere using a triple integral, we first need to define the sphere using equations for its boundaries. Then, we set up the triple integral by breaking down the volume into infinitesimal slices and integrating over each slice. The result will be the volume of the entire sphere.

What are the limits used in a triple integral for a sphere?

The limits used in a triple integral for a sphere depend on the type of coordinate system being used. In Cartesian coordinates, the limits are typically -r to r for all three variables, where r is the radius of the sphere. In spherical coordinates, the limits may vary depending on the orientation of the sphere.

Can a triple integral be used to find the volume of any three-dimensional shape?

Yes, a triple integral can be used to find the volume of any three-dimensional shape. However, the boundaries and limits may vary depending on the shape and the coordinate system being used.

How do you set up a triple integral for a sphere using spherical coordinates?

To set up a triple integral for a sphere using spherical coordinates, we first need to define the sphere using equations for its boundaries in terms of the spherical coordinates (ρ, θ, φ). Then, we set up the integral by integrating over each variable with the appropriate limits. The resulting integral will give us the volume of the sphere.

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