EX of area of a circleIn a circle k(O,r) we have fixed point A, point B lands randomly on the circle. Find EX of the area of the triangle AOBWhat I have tried: since the area of the triangle will be 1 2 ∗ r 2 ( s i n ( α ) ) where α is from 0 to 360 degrees then we have to find the integral ∫ 0 2 π 1 2 ∗ α ∗ r 2 ( s i n ( α ) ) d α .

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EX of area of a circleIn a circle k(O,r) we have fixed point A, point B lands randomly on the circle. Find EX of the area of the triangle AOBWhat I have tried: since the area of the triangle will be       1    2    ∗      r    2    (  s  i  n  (  α  )  ) where   α is from 0 to 360 degrees then we have to find the integral             ∫      0              2        π                    1      2        ∗    α    ∗          r      2        (    s    i    n    (    α    )    )        d    α  .

Kaylee Evans · Accepted Answer

Step 1The area of a triangle is absinC.Consider the scenario where O is the origin, and A is fixed to the positive x axis, from 0 to       r    1  .Then, in polar coordinates, let B be   (      r    2    ,  θ  ), where   θ ranges from   −  π to   π.The area of this triangle is             1      2              r      1              r      2              |        θ          |      .So that means the average value of the integral is             2              2        π                  r          2                            ∫      0              π                    ∫      0      r              ∫      0      r              1      2              r      1              r      2        sin    ⁡    (    θ    )        d          r      1            d          r      2            d    θ  Step 2Notice that I made the integral symmetric by noting the area is the same for   θ and   −  θ, hence the last bound only ranging from 0 to   π, and a corresponding 2 in the numerator.In your case, which I just realized, a is fixed. Eliminate       r    1   from the integral, and now you just have             2              2        π        r                    ∫      0              π                    ∫      0      r              1      2              r      1              r      2        sin    ⁡    (    θ    )        d          r      2            d    θ

In a circle k(O,r) we have fixed point A, point B lands randomly on the circle. Find EX of the area of the triangle AOB

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