|Sep26-06, 09:59 PM||#1|
Some concepts regarding area that I need help understanding
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?
|Sep27-06, 02:45 PM||#2|
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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