1. What is Spherical Integral?

A spherical integral is a triple integral computed in spherical coordinates, typically used for problems with spherical symmetry. The volume element in spherical coordinates is \( r^2 \sin\theta \, dr \, d\theta \, d\phi \).

2. How Does the Calculator Work?

The calculator uses numerical integration to approximate:

Where:

• \( f(r,\theta,\phi) \) — Function to integrate
• \( r \) — Radial distance
• \( \theta \) — Polar angle (from z-axis)
• \( \phi \) — Azimuthal angle (in xy-plane)
• \( r^2 \sin\theta \) — Spherical volume element

Explanation: The calculator divides the integration domain into small elements and sums the contributions from each element.

3. Importance of Spherical Integration

Details: Spherical integrals are essential in physics and engineering for solving problems involving spherical symmetry, such as gravitational fields, electromagnetic fields, and quantum mechanics.

4. Using the Calculator

Tips: Enter the function expression in terms of r, θ, and φ. Specify the integration limits for each variable. Use standard mathematical notation.

5. Frequently Asked Questions (FAQ)

Q1: What coordinate system is used?
A: This calculator uses spherical coordinates (r, θ, φ) where r is radius, θ is polar angle, and φ is azimuthal angle.

Q2: How accurate is the numerical integration?
A: Accuracy depends on the number of integration steps. More steps give better accuracy but require more computation time.

Q3: What functions can I integrate?
A: You can integrate any function expressible in terms of r, θ, and φ. Common examples include r², sinθ, and combinations thereof.

Q4: Are there limitations to this method?
A: Numerical integration may struggle with discontinuous functions or functions with singularities within the integration domain.

Q5: When should I use spherical coordinates?
A: Use spherical coordinates when your problem has spherical symmetry, such as when integrating over spheres or spherical volumes.