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Cylindrical coordinates extend the idea of two-dimensional polar coordinates into three dimensions by introducing an additional coordinate for height, often denoted as \( z \). These coordinates are expressed as \((r, \theta, z)\):

• \( r \) is the radial distance from the origin to the point's projection onto the \( xy \)-plane, calculated as \( r = \sqrt{x^2 + y^2} \).
• \( \theta \) is the angle between the positive \( x \)-axis and the line from the origin to the projection of the point onto the \( xy \)-plane.
• \( z \) is the height of the point above or below the \( xy \)-plane, the same as in Cartesian coordinates.

The conversion between Cartesian (\( x, y, z \)) and cylindrical coordinates is:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( z = z \)

For the given problem of finding the volume of a sphere, transforming into cylindrical coordinates makes sense because the equation of the sphere simplifies to \( r^2 + z^2 = a^2 \), which is easier to work with when setting up and evaluating integrals.

Spherical coordinates are a system that extends the concept of polar coordinates into three dimensions using three parameters: radius, polar angle, and azimuthal angle. In spherical coordinates, a point is represented by \((\rho, \phi, \theta)\):

• \( \rho \) is the radial distance from the origin to the point, equivalent to the radius. In the case of a sphere of radius \( a \), \( \rho \) equals \( a \).
• \( \phi \) is the polar angle measured from the positive \( z \)-axis downwards, ranging from 0 to \( \pi \).
• \( \theta \) is the azimuthal angle in the \( xy \)-plane, similar to \( \theta \) in cylindrical coordinates, ranging from 0 to \( 2\pi \).

Coordinates are converted as follows:

• \( x = \rho \sin(\phi) \cos(\theta) \)
• \( y = \rho \sin(\phi) \sin(\theta) \)
• \( z = \rho \cos(\phi) \)

In this context, using spherical coordinates to calculate the volume of a sphere is typically more straightforward because the limits of integration are constants and the volume integral simplifies directly to the formula for the volume of a sphere.

Triple integration is an extension of double integration to find the volume under a surface in three-dimensional space. It calculates the integral over a three-dimensional region, often involving a volume element. For the sphere, the volume element differs based on the coordinate system:

• In cylindrical coordinates, the volume element is \( dV = r \, dr \, d\theta \, dz \).
• In spherical coordinates, the volume element is \( dV = \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta \).

Each element expresses how volume increments vary based on the system used. When setting up the integral, the limits of integration represent the bounds of the region where the volume is determined. Evaluating triple integrals requires performing integration individually with respect to each variable, from the innermost to the outermost integral. This ensures an accurate computation of volume for the specified region or object. The process allows for practical applications across physics and engineering, such as calculating the volume of complex shapes that cannot be easily measured using simple geometric formulas.