General question about double and triple integral

[kevinf]

i am not using the template because it doesn't really apply to my question. can anyone explain to me when to use double integral or triple integral for volume? what clues should i look for in the question? also i am still unsure when are polar coordinates used as opposed to cylindrical and spherical.

 

[lanedance]

the general answer is whatever will make the problem easiest

I'll answer the 2nd bit first
look at the symmetry of your question, which coordinates will make the boundary of integration the simplest eg. for a sphere pick spherical coordinates... its more difficult to define the boundaries of a sphere in cartesian coords where you only know:

x^2 + y^2 + z^2 = 1 defines the boundary, where as in spherical coords all you need is r=1

Then once cordinates system is chosen write down an infintesiaml volume element, for cartesian corodinate system this is

dV = dxdydz
and a general volume integral is a triple intergral

If there is a symmetry to the problem it may be possible to write the volume in terms of less infinitesimals, eg. for a volume of a curve revolved around the z axis we can write the infintesiaml volume element as a the volume of a infinitesimally thin disk of radius based on z

dV = pi.r(z)^2.dz

so the general answer is to exploit symmetres as much as possible to make the integral as easy as possible

the other thing to remember is every variable in your integrand should be written in terms of the integration varibales

 

[kevinf]

when you typed r you mean rho right not radius for polar coordinates? when would you use polar coordinates though

 

[lanedance]

whenever the problem merits it... Every problem is different...

say you were asked to find the area of a circle by intergal, polar is easier than cartesian, but both will give the ssame answer if done correctly