Why Does My Integration Over a Sphere Give Incorrect Results?

In summary, the conversation discusses the calculation of the volume of a sphere using spherical coordinates. The correct volume element to use is ##dV = r^2 \sin(\theta) dr \, d\theta\, d\phi##. The individual steps in the integration process are also mentioned.
Homework Statement
Integrate 3 over a sphere with radius R.
Relevant Equations
Polar coordinates
##\iiint 3 dr d\rho d\phi##
The volume of a sphere is ##4\pi /3 r^3## so naturally the answer is ##4 \pi R^3##
But when I integrate I do:

##3 \iint r |_0^R d\rho d\phi##

##3R \int \rho |_0^{2\pi} d\phi##

##6R\pi * \phi |_0^\pi = 6R\pi^2##

What am I doing wrong?

What is the actual homework statement? Please reproduce the full statement exactly as given.

You have also used the wrong volume element. The volume element in spherical coordinates is ##dV = r^2 \sin(\theta) dr \, d\theta\, d\phi##.

Orodruin said:
What is the actual homework statement? Please reproduce the full statement exactly as given.

You have also used the wrong volume element. The volume element in spherical coordinates is ##dV = r^2 \sin(\theta) dr \, d\theta\, d\phi##.
it wasn't a homework but I keep being forced to post it in homework section by mods.

Eitherway thanks, I forgot to add ##r^2sin(\theta)## when converting from normal coordinates.

1. Why can't we integrate over a sphere?

Integrating over a sphere is not possible because the surface area of a sphere is infinite, making the integration process impossible to complete.

2. Can we approximate the integration over a sphere?

Yes, we can approximate the integration over a sphere by dividing the sphere into smaller sections and integrating over each section separately. However, this method may not be accurate for complex functions or large spheres.

3. Are there any alternative methods to integrate over a sphere?

Yes, instead of integrating over the entire sphere, we can integrate over a portion of the sphere, such as a hemisphere or a circular disk on the sphere's surface. This allows for a finite integration process.

4. What are some applications of integrating over a sphere?

Integrating over a sphere is commonly used in physics and engineering to calculate quantities such as electric field, gravitational force, and heat transfer over a spherical object.

5. Is there a specific formula for integrating over a sphere?

There is no one specific formula for integrating over a sphere, as it depends on the specific function being integrated and the limits of integration. However, there are general formulas and techniques that can be applied, such as spherical coordinates and numerical integration methods.

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