Parametric Integral: Understanding Math Problem

[sponsoredwalk]

Hi, I'm having some trouble understanding what's going on when integrating
the region M on page 10 of http://www.math.boun.edu.tr/instructors/ozturk/eskiders/fall04math488/bachman.pdf" , It may just be the language.

ƒ : ℝ² → ℝ defined by (x,y) ↦ z = ƒ(x,y) = y² is the function we're
integrating over the top half of the unit circle.

1: I think what he's trying to communicate in this derivation is the
standard double integral, $\int \ \int_M \ f(x,y) \ dA \ = \ \int \ \int_M \ y^2 \ dy \ dx$.
Is that correct? You'll notice he jumps straight into his paramaterization
but would what I've just done here be right?

2: If so then would the bounds on the integral become:

$\int \ \int_M \ f(x,y) \ dA \ = \ \int_{-1}^1 \ \int_0^{( \sqrt{1 - x^2})} \ y^2 \ dy \ dx$ ?

3: If that is correct then I think it would explain why the author chose to
set up a paramaterization of the region M. When he goes on to show that
the unit circle can be paramaterized in different ways it reduces a double
integral to a single integral & is just easier. Is that why?

4: I've never seen anyone paramaterize double integrals in the way he
does, could you recommend some reading material that explains what he
is doing as I can't seem to find any myself.

I have more questions, mainly to do with pages 11-14 where, I think, he is
deriving differential forms (in my meagre estimation) but I'll hold off for
now, thanks for any assistance!

 

[LCKurtz]

Hi, I'm having some trouble understanding what's going on when integrating
the region M on page 10 of http://www.math.boun.edu.tr/instructors/ozturk/eskiders/fall04math488/bachman.pdf" , It may just be the language.

ƒ : ℝ² → ℝ defined by (x,y) ↦ z = ƒ(x,y) = y² is the function we're
integrating over the top half of the unit circle.

1: I think what he's trying to communicate in this derivation is the
standard double integral, $\int \ \int_M \ f(x,y) \ dA \ = \ \int \ \int_M \ y^2 \ dy \ dx$.
Is that correct? You'll notice he jumps straight into his paramaterization
but would what I've just done here be right?

2: If so then would the bounds on the integral become:

$\int \ \int_M \ f(x,y) \ dA \ = \ \int_{-1}^1 \ \int_0^{( \sqrt{1 - x^2})} \ y^2 \ dy \ dx$ ?

No, that isn't what he is doing on page 10. He is doing a single variable integral along the curved top arc. He doesn't up the dimension until page 14.

Disclaimer: Don't expect too much more from me on this topic; I'm not a differential forms kind of guy.