Coordinate system solids

Three solid figures that could be defined in cylindrical or spherical coordinates. In cylindrical coordinates (where r is distance to the z-axis), the figures can be described by

Region 1: 0 < z < 1, z^2 + r^2 < 2
Region 2: 0 < z < r, z^2 + r^2 < 2
Region 3: 0 < z, 1 < r, z^2 + r^2 < 2

It is useful for a multivariable calculus student to think about how to calculate triple integrals over these three regions using cylindrical or spherical coordinates.

This model was designed as part of the Manipulative Calculus project at Harvard University.