To find the centroid of the region bounded by the curves y = x³, y = 10, and y = 0, we first need to determine the points of intersection of these curves.
The equation y = x³ intersects y = 10 when:
x³ = 10
Taking the cube root of both sides gives us:
x = 10^(1/3) ≈ 2.154
Thus, the bounds for our integration are x = 0 (where y = 0 intersects) and x = 10^(1/3).
Next, we compute the area of the region:
Area A can be calculated using the integral:
A = ∫[0 to 10^(1/3)] (10 – x³) dx
Evaluating this integral:
A = [10x – (x^4)/4] from 0 to 10^(1/3)
A = 10 * (10^(1/3)) – (1/4) * (10^(4/3))
Once we find the area, we can compute the centroid (x̄, ȳ) using:
x̄ = (1/A) * ∫[0 to 10^(1/3)] x(10 – x³) dx
ȳ = (1/(2A)) * ∫[0 to 10^(1/3)] (10 – x³)² dx
After solving these integrals, we extract the values for x̄ and ȳ, providing the coordinates of the centroid of the given region bounded by the curves.
The detailed calculation will result in specific numerical values for the centroid’s coordinates.