# No AM-GM here!

Calculus Level 5

$\large \displaystyle\int _{ 0 }^{ 2\pi }{ \sqrt { { a }^{ 2 }\sin ^{ 2 }{ x } +9\cos ^{ 2 }{ x } } } \, dx$

Over all constant real values of $a$, what is the minimial value of the above expression?

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