Final answer:
To find the area of the region bounded by the curves x=2y and x=y^{3}-y, we need to find the points of intersection and integrate the area between these points.
Step-by-step explanation:
To find the area of the region bounded by the curves x=2y and x=y^{3}-y, we need to find the points of intersection and integrate the area between these points. First, set the two equations equal to each other and solve for y:
2y = y^{3}-y
y^{3}-3y = 0
y(y-√3)(y+√3) = 0
So the three possible values for y are 0, √3, and -√3. Next, we determine the x-coordinates of the points of intersection by substituting these y-values into the original equations. The x-coordinates are 0, 2√3, and -2√3.
Next, we integrate the areas between the curves. The area between the curves when y∈[0,√3] is given by the integral of (y^{3}-y)dx from x=0 to x=2√3. The area between the curves when y∈[-√3,0] is given by the integral of (2y)dx from x=-2√3 to x=0. Finally, we sum these two areas to find the total area bounded by the curves.