Eroxl's Notes
Evaluating Triple Integrals in Cylindrical Coordinates

Evaluating double integrals in cylindrical coordinates is useful when the region of integration is more naturally described in cylindrical form, such as cylinders or objects with only one radial symmetry.

This method transforms the integral from Cartesian coordinates to cylindrical coordinates .

Transformation

The relationship between Cartesian and cylindrical coordinates is given by:

where , , and .

The area element in polar coordinates is . The additional factor of arrises from the Jacobian of the transformation from Cartesian to polar coordinates.

Triple Integral Formula

To evaluate over a region in cylindrical coordinates:

  1. Express in terms of , , and .
  2. Determine the limits for , , that describe .
  3. Use the formula: