13.7no7

I don't know why its coming out wrong!

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A few reasons.  First the vector element of area dA here is what our book calls dS, and not what the book calls dA--that is dA=(r_u x r_v )dudv, or since your parameters are called  r, θ  
dA=(r_r x r_θ )drdθ.  Second,  r, θ here are just parameters, not polar coordinates: you only have drdθ for your area element in the parameter plane, not rdrdθ (in fact I guess that the whole point of this problem is to fool you into making the mistake you did make, so that you can learn not to make it).  Finally, in your integral ∫∫_S F.dA it looks like you dropped the 9 from your F and also did the dθ integral incorrectly: the answer is not 2π because the F . (r_r x r_θ ) actually not independent of θ.