Surface Area of of an area - parametric surface

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Homework Help Overview

The discussion revolves around finding the surface area of the portion of a sphere defined by the equation x² + y² + z² = 1 that lies above the cone described by z = √(x² + y²). Participants are exploring various methods to approach this problem, including the use of cylindrical and spherical coordinates.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss different equations for surface area and question which to use in the context of the sphere and cone. There are attempts to clarify the relationship between the equations and the geometric shapes involved. Some participants suggest using polar or cylindrical coordinates, while others express confusion about the integration limits and the setup of the problem.

Discussion Status

The discussion is ongoing, with participants providing various insights and suggestions regarding the use of different coordinate systems. Some have offered guidance on how to set up integrals, while others are still grappling with the correct approach and expressing uncertainty about their calculations.

Contextual Notes

Participants are navigating the complexities of integrating over a surface area defined by the intersection of a sphere and a cone, with some expressing difficulty in transitioning between coordinate systems and understanding the implications of their choices on the integration process.

  • #31
I got a + sign in there. (-1)^2=(+1). You can still simplify some more.
 
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  • #32
Woops, your right:

A(S) = \int \int \sqrt{\frac{1}{(1-r^{2})} dA
 
  • #33
Ok, now you want to integrate over the values of x and y that will cover the spherical cap. That's a circular region, right? That's a good reason to switch to polar coordinates. What is dA in polar coordinates?
 
  • #34
r^2 sin (phi) dr dtheta dphi
 
  • #35
A is a region in the x-y plane. Use polar coordinates. Not spherical.
 

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