Surface area of cap using integrals

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SUMMARY

The discussion focuses on calculating the surface area of a spherical cap defined by the sphere equation x² + y² + z² = 2 and the cone equation z = sqrt(x² + y²). The correct surface area is determined to be 2π(2 - sqrt(2)). The solution involves using spherical coordinates, specifically the relationships z = r cos(θ) and r² = x² + y² + z², to simplify the integration process. The method of finding the normal vector and applying the cosine formula is also highlighted as a crucial step in the solution.

PREREQUISITES
  • Understanding of spherical coordinates
  • Familiarity with surface area integrals
  • Knowledge of vector calculus, specifically normal vectors
  • Proficiency in trigonometric identities and transformations
NEXT STEPS
  • Study the application of spherical coordinates in surface area calculations
  • Learn about the derivation and use of normal vectors in calculus
  • Explore integration techniques for surface area using double integrals
  • Review examples of surface area problems involving cones and spheres
USEFUL FOR

Students studying calculus, particularly those focusing on multivariable calculus and surface integrals, as well as educators looking for examples of spherical coordinate applications in geometric problems.

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Homework Statement


The question asks,

"Find the surface area of the cap cut from the sphere x^2+y^2+z^2=2 by the cone z = sqrt(x^2+y^2)" The answer should be 2pi(2-sqrt(2))

My main problem is not knowing how to get started.

Homework Equations



With the example given, it seems we need to find cos(v) first using the equation cos(v) = n*.k/|n|.

The Attempt at a Solution



I found the normal line to be 2xi+2yj+2zk. Using the above formula, I eventually reached the conclusion that z/sqrt(r^2+z^2). I don't know how to use this in an integral and it doesn't follow the example our professor gave us either. Can anyone help?
 
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Tame this problem by writing:

z = r cos (theta) (here theta is the zenith)

and

r^2 = x^2 + y^2 + z^2.

Work in spherical coordinates. It'll be that much easier.
 

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