Denotes the greatest integer function, then the value of A is:

To find the area bounded by the curves, we need to find the intersection points first.

For y = sin^(-1)(sin(x)), the range of y is [-π/2, π/2]. This means that the curve will only exist in this range.

For y = [|x|], the greatest integer function will return the integer part of x. So for positive x, y = x, and for negative x, y = -x.

To find the intersection points, we need to find when sin^(-1)(sin(x)) = x and when sin^(-1)(sin(x)) = -x.

When sin^(-1)(sin(x)) = x, we can simplify it as sin(x) = sin(x). This is true for all values of x, so the curve y = sin^(-1)(sin(x)) intersects the line y = x for all values of x.

When sin^(-1)(sin(x)) = -x, we can simplify it as sin(x) = -sin(x). This is only true when sin(x) = 0, which happens at x = 0, π, 2π, etc.

So the intersection points are (0, 0), (π, π), (2π, 2π), etc.

To find the area bounded by the curves, we need to find the areas between these intersection points.

The area between (0, 0) and (π, π) is given by the integral of sin^(-1)(sin(x)) - x from x = 0 to x = π.

∫(sin^(-1)(sin(x)) - x) dx from x = 0 to x = π

Using integration by parts, let u = sin^(-1)(sin(x)) and dv = dx, then du = (cos(x))/(√(1 - sin^2(x))) dx and v = x.

The integral becomes:

[sin^(-1)(sin(x))x] from x = 0 to x = π - ∫[(cos(x))/(√(1 - sin^2(x))))x dx from x = 0 to x = π

The integral of (cos(x))/(√(1 - sin^2(x)))) is a bit complicated, but it simplifies to -π/2.

So the area between (0, 0) and (π, π) is:

[sin^(-1)(sin(x))x - πx/2] from x = 0 to x = π

= (ππ - π(0))/2 - (0(0) - π(0))/2

= π^2/2

The area between (π, π) and (2π, 2π) is also π^2/2, since the curve y = sin^(-1)(sin(x)) intersects the line y = x for all values of x.

So the total area bounded by the curve y = sin^(-1)(sin(x)), y = [|x|] from x = 0 to x = π is:

π^2/2 + π^2/2 = π^2 sq. units.

Therefore, the value of A is π^2 sq. units.

Y = {[|x|]} = 0