Area of an ellipse and similar problems

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SUMMARY

The area of an ellipse can be accurately calculated by breaking it into infinitesimally small triangles and integrating, as discussed in the forum. The alternative method of using infinitesimally small circle segments raises questions about its validity, particularly in terms of integration in polar coordinates. The discussion emphasizes the importance of choosing the correct method for integration to ensure accurate results, highlighting that not all approaches yield the same outcome. Understanding the governing principles behind these methods is crucial for solving similar geometric problems.

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  • Understanding of calculus, specifically integration techniques.
  • Familiarity with polar coordinates and their applications in geometry.
  • Knowledge of geometric properties of ellipses and triangles.
  • Basic principles of mathematical reasoning and proof construction.
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  • Explore integration techniques for calculating areas of geometric shapes.
  • Learn about polar coordinate integration and its applications in various problems.
  • Investigate the properties and equations governing ellipses and their areas.
  • Study the comparison of different integration methods and their effectiveness in geometry.
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Mathematicians, students studying calculus, educators teaching geometry, and anyone interested in advanced integration techniques for geometric shapes.

Gauss M.D.
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To find the area, you break the ellipse into infinitesimaly small triangles and integrate.

But why? Why not break it up into infinitesimaly small circle segments and calculate it through circumference instead?

There are other problems regarding integration of geometric objects that has me wondering the same thing. Integrating a function in polar coordinates for example. It seems to me that the triangle method and the circle method has equal logical merit. But one of them produces the wrong result. What's governing which one to choose?
 
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Could you put your circle segment idea in mathematical terms? I am sure that it works if you do it right.
 

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