14.13
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.
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.
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.
From Chapter 14:
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