Pauls Online Notes : Calculus III - Triple Integrals in Spherical Coordinates
So, looking at example 2, particularly at
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...
What I'm confused about is
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?