Sketching Solids: Overcome Inequalities & Cosecant

[kidzonety]

I always have trouble sketching regions and solids.
For example, if one solid described by the inequialities (in spherical coordinates) where ρ <=2, and ρ <= csc(φ)
I don't get how to interpret the cosecant part. Thanks.

 

[slider142]

What surface is described by [itex]\rho = \csc\varphi[/itex]? This is part of the boundary of the region described.
If you are unfamiliar with the geometry of cosecant, have a look at the picture on the page http://en.wikipedia.org/wiki/Cosecant#Reciprocal_functions and try graphing r = csc(t) in polar coordinates (r, t) first. It is a very simple graph.

 

[LCKurtz]

I always have trouble sketching regions and solids.
For example, if one solid described by the inequialities (in spherical coordinates) where ρ <=2, and ρ <= csc(φ)
I don't get how to interpret the cosecant part. Thanks.

Sometimes changing to coordinate system gives insight. Here the surface of the solid is ρ = csc(φ). If r is usual the cylindrical coordinate you have ρ = ρ/r or r = 1. Can you describe the solid given by r ≤ 1 with ρ ≤ 2?