Using a Surface Integral for Mathematical Analysis of the Area of an Island

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SUMMARY

The discussion focuses on using a surface integral to analyze the area of an island shaped like an upside-down paraboloid. Participants emphasize the importance of correctly setting the limits of integration for the variable r, stating that they should be reversed to ensure a positive result. The link to the Wikipedia page on spherical caps is provided as a reference for understanding the geometric implications. Overall, the correct application of surface integrals is crucial for accurate mathematical analysis in this context.

PREREQUISITES
  • Understanding of surface integrals in calculus
  • Familiarity with the geometry of paraboloids
  • Knowledge of integration limits and their significance
  • Basic concepts of mathematical analysis
NEXT STEPS
  • Study the properties of surface integrals in calculus
  • Research the geometric characteristics of paraboloids
  • Learn about the implications of integration limits in calculus
  • Explore applications of surface integrals in real-world scenarios
USEFUL FOR

Mathematicians, engineering students, and anyone involved in geometric analysis or calculus who seeks to understand the application of surface integrals in determining areas of complex shapes.

daphnelee-mh
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Homework Statement
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Relevant Equations
A=sqrt[1+(dz/dx)^2+(dz/dy)^2]dA
1593786915216.png

I am not clearly understand what the question requests for, is it okay to continue doing like this ? Kindly advise, thanks
 
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The shape of the island is an upside-down paraboloid. The integral looks like everything is OK except the ##r## limits should be reversed. You want to integrate from the smallest to largest values of ##r## to get a positive answer.
 
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LCKurtz said:
The shape of the island is an upside-down paraboloid. The integral looks like everything is OK except the ##r## limits should be reversed. You want to integrate from the smallest to largest values of ##r## to get a positive answer.
I believe you're right. I looked at it wrong.
 

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