Area Of An Infinite Batwing

FOR n = 0 TO INFINITY PRINT [(1/2)^n]sin[(2^n)x]

It's an infinitude of sine waves. The first function, sin x, runs from 0 to π. We take a bite out of the area above the x-axis. We do this with a sine wave having half the amplitude and twice the frequency. 

Create a third sine wave, half the amplitude and twice the frequency of the previous. Use it to take a bite out of the remaining area.

Repeat.

Forever.

The remaining shaded area looks like a batwing. What is the size of the batwing? ... https://www.youtube.com/watch?v=JZ7SZVVVSIk