I am trying to find a function prescribed in polar coordinates r = f ( θ ) that maximizes the following quantity ∫ 0 2 π r 3 cos ⁡ θ d θ ∫ 0 2 π r 4 d θ subject to the constraint r ≤ R ∀ θ ∈ [ 0 , 2 π ).I don't even know where to begin to be honest. I went through a couple of books on calculus of variations. I haven't seen anything that comes close to the problem I have at hand.

Question

I am trying to find a function prescribed in polar coordinates   r  =  f  (  θ  ) that maximizes the following quantity                    ∫        0                  2          π                            r        3            cos      ⁡      θ            d      θ                      ∫        0                  2          π                            r        4                  d      θ      subject to the constraint   r  ≤  R    ∀  θ  ∈  [  0  ,  2  π  ).I don&#039;t even know where to begin to be honest. I went through a couple of books on calculus of variations. I haven&#039;t seen anything that comes close to the problem I have at hand.

iceniessyoy · Accepted Answer

Clearly the maximum is   +  ∞. Consider for example   r  =  f  (  θ  )  =  ϵ      |    cos  ⁡  (  θ      /    2  )      |  . Then                    ∫        0                  2          π                            r        3            cos      ⁡      θ      d      θ                      ∫        0                  2          π                            r        4            d      θ        =      32          15      π      ϵ      and this tends to   +  ∞ as   ϵ goes to       0    +  .

I am trying to find a function prescribed in polar coordinates r = f ( θ<!--

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