Triple Integrals in Cylindrical Coordinates

In a cylindrical coordinate system, a point P in three-dimensional space is described using the ordered triple r, theta, and z.

Here, z gives the height above the xy-plane, r represents the radial distance from the z-axis, and theta is the angle measured from the positive x-axis in the xy-plane.

To calculate the volume for regions with circular symmetry, x and y are converted to r and theta. The region is then divided into small cylindrical wedges to reflect this geometry.

Each wedge has three small dimensions consisting of a change in height, a change in radius, and an arc length determined by the change in angle.

Multiplying these dimensions gives the volume of a single wedge.

Summing the volumes of all wedges gives the Riemann sum.

As the wedges become infinitely small, this Riemann sum approaches a triple integral.

A triple integral in cylindrical coordinates can be used to find the volume of a cylindrical-shaped storage tank. Here, the limits define the height from the base to the top, the radial distance to the outer wall, and the full rotation from 0 to 2pi. Solving this gives the known formula for a cylindrical storage tank.