Filtered Curve vs. Original (Part 2)

Two discrete curves are defined as follows:

$\Delta x = \frac{\pi}{1000} \\ \alpha = 0.995 \\ k = 0,1,2,3 .... 2000 \\ x_k = k \, \Delta x \\ y_k = sin(x_k) \\ y'_k = \alpha \, y'_{k-1} + (1 - \alpha) y_k \\ y'_0 = 0$

Both curves are plotted on the diagram as smooth lines for display purposes. The $k$ subscript denotes the present value of the variable, and the $k-1$ subscript denotes the previous value of the variable. The quantity $y'$ is the result when $y$ is passed through an infinite-impulse-response filter.

Define the infinitesimal area and total area between curves as follows:

$\Delta A_k = | y_k - y'_k | \, \Delta x \\ A = \Sigma_{k = 0}^{k = 2000} \, \Delta A_k$

What is the value of $A$ ?