First, we need to find the x-coordinates of the intersection points of the curves:
sinπx = 4x - 1
πx = arcsin(4x - 1)
x = arcsin(4x - 1)/π
The two curves intersect at two points, one of which is at x=0, the other one can be found numerically (e.g. using a graphing calculator) to be at approximately x=0.72.
Therefore, the area of the region can be found by dividing it into two triangles and one curved region:
Area = A1 + A2 + A3
where A1 is the area of the triangle bounded by the x-axis and the vertical lines x=0 and x=0.72, A2 is the area of the triangle bounded by the x-axis and the vertical lines x=0.72 and x=1, and A3 is the area of the curved region between the two vertical lines.
A1 = (1/2) * 0.72 * sin(π*0.72) ≈ 0.369
A2 = (1/2) * (1-0.72) * (4*0.72 - 1) = 0.4032
A3 = ∫[0.72,1] sin(πx) dx ≈ 0.4343
Therefore, the total area of the region is:
Area = A1 + A2 + A3 ≈ 1.2065
Answer: The area of the region bounded by the curves y=sinπx, y=4x-1, and the x-axis is approximately 1.2065 square units.