Confused conceptually about spherical integration and the length of the sphere's r

Pauls Online Notes : Calculus III - Triple Integrals in Spherical Coordinates

So, looking at example 2, particularly at

http://tutorial.math.lamar.edu/Class...s/eq0028MP.gif

I solved a similar problem using cylindrical coordinates, and decided to resolve it using spherical coordinates for practice and ended up confusing myself horribly. The way I understand it, the intersection of the sphere and cone create a projected region that is the domain of x and y's that give you every z value that make the surface of the cone. However, when you set up limits of integration, you are creating a smaller domain and evaluating the z values of that domain. In a 2d representation...

http://i.imgur.com/QOVK85g.png

What I'm confused about is

http://tutorial.math.lamar.edu/Class...s/eq0028MP.gif

judging from this diagram, it seems like the size of the base no longer matters? For example, I could change the limits of integration so that it was a half disk of radius 4 and I would get the exact same integral for spherical coordinates when I thought that you should receive a larger number for having a larger projected region. I originally thought that what would change is the length of the sphere's radius, but in Paul's notes he simply finds the length of the sphere's radius with x^2+y^2+z^2=18.

Does this question make sense?

Re: Confused conceptually about spherical integration and the length of the sphere's

Hey TheBlackCat.

Triple integrals can be confusing but apart from the mathematical substitutions for limits and integrands, the best advice I can give is to draw a diagram for each substitution. In a 2D integral you have a plane and in a 3D integral you have a volume.

However you can also look at each 2D integral (there will be 3 pairs XY XZ and YZ) and use this to visualize what is going on each pair and then use that intuition to see what is happening in the 3D space. (This is also useful if you have to get intuition for 4D or higher space - You just slice up the components and integrate them together as intuitively as you can).

Do you want us here to help you with the algebraic derivation or the intuitive geometric concepts?

Re: Confused conceptually about spherical integration and the length of the sphere's

I worked through it and I think I actually understand a little better. From my understanding, the length of rho should change depending on the size of the projected region, but the books tend to use examples that don't need this recalculation. Thank you for your reply though!