Some concepts regarding area that I understanding

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SUMMARY

This discussion focuses on the relationship between 3D volume formulas and their corresponding 2D area formulas, specifically through the lens of calculus and integration. The derivation of the volume formula for a cone from the area formula of a circle is highlighted, utilizing a 3D coordinate system where the cone's vertex is at the origin. The formula for the volume of a cone is established as V = (πR²h)/3, derived by integrating the differential volume πr²dx from 0 to h, where r is defined as kx with k = R/h.

PREREQUISITES
  • Understanding of calculus, specifically integration techniques
  • Familiarity with 3D coordinate systems
  • Knowledge of geometric shapes, particularly circles and cones
  • Basic proficiency in mathematical notation and formulas
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  • Study the principles of calculus integration in depth
  • Explore the derivation of volume formulas for other geometric shapes
  • Learn about the applications of 3D coordinate systems in geometry
  • Investigate the relationship between 2D and 3D geometric properties
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Students and educators in mathematics, particularly those focusing on geometry and calculus, as well as professionals involved in fields requiring spatial reasoning and volume calculations.

fwb
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Most 3d volume formulas look similar to their corresponding 2d area formulas for a particular shape. How does calculus, specifically integration, relate the two sets of formulas?

For example: how is the formula of a cone derived from the formula of a circle?
 
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One method: 3d coordinate system - put cone point at origin and cone axis along positive x axis. Let h be height of cone and R=radius of base. Let k=R/h. The radius of the cone circle defined by a plane at point x perpendicular to the x-axis is given by r=kx. The differential of volume at a point x is then pi*r2dx. Integrate from 0 to h and the volume will be pi*k2h3/3=pi*R2h/3.

There may be neater ways, but you can see the point.
 

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