How do I integrate a hemisphere using spherical coordinates?

[bobie]

Integrating a hemisphere

Homework Statement

I am self-teaching calculus.
I'd like to integrate a hemisphere G (http://en.wikipedia.org/wiki/Spherical_cap), where the points of each circle have a given property (k=100x/(√20x)³
the radius of G r = 10, A = 628.3, h = x
the top point is A (x=0), a = √20x-x², and the line AP joining a to a point P is √x²+20x-x² = √20 x

I learned to find an approximation by discrete summation, finding the value of y for every slice of G of height 1, (whose area is 20 pi = 62.8) (http://www.wolframalpha.com/input/?i=(sum+[+100x/sqrt(20x)^3,{x+,1,10}])+*20pi) y =352.7

Now, I'd like to find the exact solution, integrating the product of y by the area of a single circle
h , I suppose x must be = h as dx must be infinitesimal, is that right?
http://www.wolframalpha.com/input/?i=integrate+y+=+100+*x/sqrt(20*x)^3+*20pi+x+from+1+to+10) y= 304
There is a 16% difference, is it too much or everything is OK?

Thanks

 

[Mogarrr]

I think this can be done by switching to spherical coordinates and doing a triple integral.

Here's a youtube link I found: .

and this is from Paul's Online Calculus Notes: http://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx.