Final answer:
To find the centroid bounded by the curves y = x⁴ and x = y⁴, we note that the region is symmetrical and the centroid lies at the midpoint (0.5, 0.5) of the intersection points due to this symmetry.
Step-by-step explanation:
To find the centroid of the region bounded by the curves y = x⁴ and x = y⁴, we first need to determine the intersection points of these curves. Equating the two equations, we have x⁴ = y⁴, which simplifies to x = y after taking the fourth root of both sides. Therefore, we have the points (0,0) and (1,1) as the limits for integration.
The centroid (α, β) has coordinates given by the formulas α = ∫ x dA / A and β = ∫ y dA / A, where A is the area of the region and dA is a differential element of area. Since we are dealing with symmetrical curves with respect to the line y=x, we can infer that the centroid lies on this line, so α = β.
Integrations to find α and β would require setting up double integrals with the bounds we found earlier. However, due to the symmetry, we know that the centroid will be at the midpoint of the intersection points, which is at (0.5, 0.5).