Area with a Twist(Corrected)

Calculus Level 5

The area bounded by \(y=f(x)\), \(x=\frac{1}{2}\), \(x=\frac{\sqrt{3}}{2}, \) and the \(x\)-axis is \(A\) square units, where \[ f(x) = x+ \dfrac{2}{3}x^3 + \dfrac{2}{3}\cdot\dfrac{4}{5}x^5 + \dfrac{2}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{7}x^7+ \cdots \] and \(|x| < 1.\)

If \(A\) can be written as \(\dfrac{\pi^a}{b} \), where \(a\) and \(b\) are positive integers, find \( a + b \).