Cartesian to cylindrical coordinates (integration question)

[Miike012]

There has been a few times when I switch from Cartesian to cylindrical coordinates to integrate I would get the wrong because I used the wrong substitution.
For instance I would use x = rcos(θ) and y = rsin(θ) where r and θ are variable when I was suppose to leave r as a constant.

Question: correct me if I am wrong, I should use x = rcosθ and y = rsinθ where r is variable if the cross section parallel to my region of integration are circles whose radius are not constant. For example: a cone.

And I would choose r to be the appropriate constant if the cross sections are circles with constant radius for example the surface x^2 + y^2 = 16 ... a cylinder.

Is there anything else I should know?

 

[LCKurtz]

A surface requires two parameters while curve requires only 1. For example, to describe the unit circle ##x^2+y^2=1## and its interior you could use ##x=r\cos\theta,\, y=r\sin\theta## where ##r## varies from ##0## to ##1## and ##\theta## varies from ##0## to ##2\pi##. If you just set ##r=1## then you just get the curve enclosing the area. If you set ##\theta = \pi/4## and let ##r## vary you get the ##45^\circ## line.