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Thread: How to calculate the area in buffon needle problem?

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    How to calculate the area in buffon needle problem?




    i am confused to compute the area in buffon needle problem, 1.could you tell me which coordinate to choose ? polar coordinate or cartesian coordinates?
    2.if polar is to be choosen ,where is the pole point?
    3.how can we calculate the area that way ?(please see the buffon.JPG)
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    Re: How to calculate the area in buffon needle problem?

    In the figure, lets call the horizontal axis to be the \theta-axis and the vertical axis to be the y axis.

    Then we have the curve y = L\sin\theta, ~~ \theta \in [0, \pi] inside the \theta-y Cartesian coordinates plane.

    To calculate the area between the curve and the \theta-axis y = 0, we have the integral

    \int_0^{\pi} (L\sin\theta - 0)d\theta

    In essence you are summing the areas of infinitely many rectangles with infinitesimally small width d\theta and the heights (L\sin\theta - 0), which is the integrand and it is varying along the curve as \theta changes.

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    Re: How to calculate the area in buffon needle problem?


    i still don't understand,the horizontal axis is a distance or a angle?
    if the horizontal axis is a distance ,the coordinate is cartesian coordinates,then you can not use y=L*sin(x)

    if the horizontal axis is a angle ,the coordinate is polar coordinates,then what is the pole point?

    in this case, you mix cartesian coordinates and polar coordinates into a nosence coordinates ??

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