Coordinate system solids

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Three distinct shapes that can be defined in terms of either cylindrical or spherical coordinates are easily identified. In cylindrical coordinates (where r represents distance to the z-axis), these shapes can be described as follows: Region 1: For every point where z is greater than 0 and less than 1, and for all points where z squared plus r squared is less than 2. Region 2: For every point where z is greater than 0 and less than r, and for all points where z squared plus r squared is less than 2. Region 3: For every point where z is greater than 0 and for all points where r is greater than 1 and z squared plus r squared is less than 2. It is beneficial for a student of multivariable calculus to consider how to calculate triple integrals over these three regions using either cylindrical or spherical coordinates. This model was designed as part of the Manipulative Calculus project at Harvard University, which aims to enhance understanding in calculus through interactive learning tools.