球面座標における三重積分

Triple integrals in spherical coordinates are ideal for objects with central symmetry.

In spherical coordinates, any point in space is described using three variables.

Here, rho represents the distance from the origin, while theta is the angle in the xy plane, measured from the x-axis, and phi is the angle measured from the positive z-axis.

These coordinates can be seamlessly used to derive the standard formula for the volume of a sphere. First, the sphere is divided into tiny spherical wedges, each defined by changes in rho, theta, and phi.

The volume of each wedge is found by multiplying its radial thickness by its two distinct circular arc lengths.

The total volume is then calculated by integrating this small wedge over the entire sphere.

To set the limits, rho ranges from 0 at the center to the outer radius R. Next, phi sweeps vertically from 0 to pi.

Finally, theta rotates a full 0 to 2pi around the z-axis to sweep out the entire three-dimensional shape.

Solving this integral with these limits gives the known formula for the volume of a sphere of radius R using spherical coordinates.