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  • Circle is special

    Shah Sharath

    Member

    Updated: Oct 25, 2024
    Views: 1.2K
    Why differentiation of area of circle is its circumference
    And vice versa while integrating ??
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Replies

  • zaveri

    Member • Mar 1, 2014

    The differentiation of area of circle is 2*pi*r dr

    Do not forget the "dr".

    whereas the circumference formula is just 2*pi*r.

    And the circumference formula cannot be integrated unless it has a "dr" along with it.
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  • Ramani Aswath

    Member • Mar 1, 2014

    If f(x) = a x^b , then df(x)/dx = abX^(b-1)
    It just happens that the area of the circle has a differential equal to the circumference as a consequence of the above.
    See what happens if you use the area of the circle as a function of the diameter.

    Area = (pi/4)D^2
    The differential wrt d = (pi/2)D, which is not the circumference.
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  • zaveri

    Member • Mar 1, 2014

    A.V.Ramani
    If f(x) = a x^b , then df(x)/dx = abX^(b-1)
    It just happens that the area of the circle has a differential equal to the circumference as a consequence of the above.
    See what happens if you use the area of the circle as a function of the diameter.

    Area = (pi/4)D^2
    The differential wrt d = (pi/2)D, which is not the circumference.
    That is some logic.
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  • Shah Sharath

    Member • Mar 4, 2014

    A.V.Ramani
    If f(x) = a x^b , then df(x)/dx = abX^(b-1)
    It just happens that the area of the circle has a differential equal to the circumference as a consequence of the above.
    See what happens if you use the area of the circle as a function of the diameter.

    Area = (pi/4)D^2
    The differential wrt d = (pi/2)D, which is not the circumference.
    A.V.Ramani
    If f(x) = a x^b , then df(x)/dx = abX^(b-1)
    It just happens that the area of the circle has a differential equal to the circumference as a consequence of the above.
    See what happens if you use the area of the circle as a function of the diameter.

    Area = (pi/4)D^2
    The differential wrt d = (pi/2)D, which is not the circumference.
    OH!!!!! that not working for diameter....
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  • Shashank Moghe

    Member • Sep 15, 2014

    Shah Sharath
    Why differentiation of area of circle is its circumference
    And vice versa while integrating ??

    First of all, let me immensely thank you for this acute observation. I have been studying Math for years and have never seen that fact standing right there in front of me. This got me thinking. And that is something I love. Apart from the rote d/dr purely mechanical explanation of why the area derivated equals circumference and the circumference integrated equals area, I wanted to have a better analytical explanation for your question.

    I hope I have made myself plenty clear. Please let me know if this clears your doubt.

    Once again, Thank you. Keep the curiosity up, challenge everything!
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  • shiwa436

    Member • Sep 16, 2014

    As from the definitions of integration and differentiation from the Wikipedia,
    Integration of perimeter 2pir with respect to its radius r gives its signed area in both the planes i.e x, y planes..

    Differential gives sensitivity of position of y axis equivalent w.r.t x axis one... i.e position of a dot on the edge along the diameter
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  • Shashank Moghe

    Member • Sep 17, 2014

    Please be more detail. Your answer is more "cryptic", please make sure it is described in the simplest of words.
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  • Ramani Aswath

    Member • Sep 17, 2014

    shiwa436
    As from the definitions of integration and differentiation from the Wikipedia,
    Integration of perimeter 2pir with respect to its radius r gives its signed area in both the planes i.e x, y planes..
    Differential gives sensitivity of position of y axis equivalent w.r.t x axis one... i.e position of a dot on the edge along the diameter
    Please study the problem in detail and refelect on the actions performed.
    There is no 'x' plane or 'y' plane. There is an x-y plane. The circle can be on any plane x-y or something else.
    In this example the differential gives the length of the perimeter, not a point on the circumference.
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