Triple Integrals in Spherical Coordinates

Triple integrals in spherical coordinates provide an efficient method for evaluating volumes over regions with central symmetry, such as spheres. Instead of describing points by rectangular coordinates, spherical coordinates use three variables: 𝜌, 𝜃, and 𝜑. Here, 𝜌 is the distance from the origin, 𝜃 is the angle in the xy-plane measured from the positive x-axis, and 𝜑 is the angle measured downward from the positive z-axis.

To derive the volume of a sphere, the solid region can be divided into small spherical wedges. Each wedge is determined by small changes in 𝜌, 𝜃, and 𝜑. Its approximate volume is found by multiplying the radial thickness by two circular arc lengths. This wedge structure accounts for how distances spread out as 𝜌 increases and as the direction changes over the sphere.

For a sphere of radius R, 𝜌 ranges from the center of the sphere to its outer surface. The angle 𝜑 sweeps vertically from the positive z-axis to the negative z-axis, covering the full height of the sphere. The angle 𝜃 rotates once around the z-axis, sweeping through the entire circular direction in the xy-plane.

The total volume is obtained by adding the volumes of all the tiny spherical wedges throughout the sphere. Because the limits cover every point inside the sphere exactly once, this triple integral gives the full volume of the solid. After evaluation, the result is the standard volume formula for a sphere of radius R.