(pi*R²)'=2*pi*R... any meaning to that?
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Is there any meaning to that fact that the driven function of a circle''s area is it''s perimiter?
Consider the question in reverse. Integrals are used for determining the area beneath a curve, yes?
What''s also cool is that a sphere''s volume (4/3 pi R^3) is the integral of it''s surface area (4 pi R^2)
This lovely piece of mathematics that you have inquired about is based on Green's Theorem, which is a special case of Stokes' Theorem.
Green's Theorem states the equivalence between a line integral around a simple closed curve C and the surface integral over the plane bounded by C . Stokes' Theorem generalises this to n -dimensional space curves.
So, according to Green's Theorem, the line integral along the curve given by the circle is equivalent to the surface integral over the region bounded by the circle, which computes the area bounded by the circle.
If you want to know more, grab an introductory text on Calculus & Analytic Geometry !
Cheers,
Timkin
Edited by - Timkin on February 17, 2002 8:54:04 PM
Green's Theorem states the equivalence between a line integral around a simple closed curve C and the surface integral over the plane bounded by C . Stokes' Theorem generalises this to n -dimensional space curves.
So, according to Green's Theorem, the line integral along the curve given by the circle is equivalent to the surface integral over the region bounded by the circle, which computes the area bounded by the circle.
If you want to know more, grab an introductory text on Calculus & Analytic Geometry !
Cheers,
Timkin
Edited by - Timkin on February 17, 2002 8:54:04 PM
The relationship between the circle and sphere area/volume and boundary is also worth remembering because it generalises to other shapes and properties.
E.g. the moment of inertia: the moment of inertia of a ring/hoop about its axis of rotation is simply mass x radius ^2, and this can be integrated to work out the moment of inertia of a disc. The moment of inertia of many other shapes can be worked out the same way.
E.g. the moment of inertia: the moment of inertia of a ring/hoop about its axis of rotation is simply mass x radius ^2, and this can be integrated to work out the moment of inertia of a disc. The moment of inertia of many other shapes can be worked out the same way.
quote:Original post by johnb
The relationship between the circle and sphere area/volume and boundary is also worth remembering because it generalises to other shapes and properties.
No offence intended, but didn''t I just say that!??? To quote myself:
quote:
Green''s Theorem states the equivalence between a line integral around a simple closed curve C and the surface integral over the plane bounded by C . Stokes'' Theorem generalises this to n-dimensional space curves.
As for the moment of inertia computation... yes, you are correct. More generally, the second moment of any density distribution can be computed by an equivalent line integral around the boundary of the distribution.
Cheers,
Timkin
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