\( \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ a{ n }^{ 2 } } =\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 4-{ x }^{ 2 } } } } } \)

Find the value of \(a\).

#### This problem is original.

#### Picture credits: *Rainbow 02* by Jerry Magnum Porsbjer, Wikipedia