Practice question · Sort into groups
Choose the natural coordinate system for integrating over each solid.
Groups: Cylindrical · Spherical · Cartesian
- A pipe segment (annular cylinder)
- A soup can (finite cylinder)
- A shoebox
- An ice-cream cone capped by a sphere chunk
- A planet (solid ball)
Hints
- Pick the system in which each boundary surface reads "coordinate = constant".
- A cone through the origin is constant in spherical, that is why the cone-plus-cap solid is a spherical box.
Show the answer
Cylindrical: A soup can (finite cylinder), A pipe segment (annular cylinder)
Spherical: A planet (solid ball), An ice-cream cone capped by a sphere chunk
Cartesian: A shoebox
Why
Every boundary that becomes "coordinate = constant" removes one hard limit. Cans and pipes are cylindrical boxes; balls and cones are spherical boxes; crates are Cartesian ones. Geometry chooses the coordinates, not habit.
Practise Polar, Cylindrical, and Spherical Coordinates
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