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)

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

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

Then we have the curve inside the Cartesian coordinates plane.

To calculate the area between the curve and the -axis , we have the integral

In essence you are summing the areas of infinitely many rectangles with infinitesimally small width and the heights , which is the integrand and it is varying along the curve as changes.

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 ??