Triple Integrals in Cylindrical Coordinates

Cylindrical coordinates describe a point in three-dimensional space using three values: radial distance, angle, and height. The height gives the position above the xy-plane, the radial distance measures how far the point is from the z-axis, and the angle describes the point’s direction from the positive x-axis in the xy-plane. This system is especially useful for regions with circular symmetry because it matches the natural geometry of cylinders, disks, and circular tanks.

To calculate volume, the region is divided into many small cylindrical wedges. Each wedge has a small height, a small radial thickness, and a small curved width along the circular direction. The curved width depends on both the radial distance and the change in angle. Therefore, wedges farther from the z-axis are wider than wedges closer to the center.

The volume of each small wedge is found by multiplying these three small dimensions. Adding the volumes of all wedges produces a Riemann sum, which approximates the total volume of the region.

As the wedges become infinitely small, the Riemann sum approaches a triple integral in cylindrical coordinates. The limits of this integral describe the vertical height, the radial distance from the central axis to the outer boundary, and the angular sweep around the z-axis.

For a cylindrical storage tank, the height limits extend from the base to the top, the radial limits extend from the center to the outer wall, and the angular limits complete one full rotation around the tank. Evaluating the corresponding triple integral gives the familiar result: the volume of a cylinder equals the area of its circular base multiplied by its height.