Find the centroid of the region bounded by the curves y = 1 and y = x^2.

Answer 1

Let find intersection points: 1 = x², x = ±1, so we have for this region: x² ≤ y ≤1 and -1 ≤ x ≤ 1.

Area of region: A = ∫_-1¹(1 - x²)dx = (x - x³/3)_-1¹ = ( 1 - 1/3) - (- 1 + 1/3) = 4/3.

Centroid C: x_C = ∫_-1¹x(1 - x²)dx/A = 0, because we have integral of odd function in symmetric limits.

y_C = 1/2∫_-1¹(1 - x⁴)dx/A = 1/2(x - x⁵/5)_-1¹/(4/3) = 1/2·8/5/(4/3) = 3/5 = 0.6

C(0, 0.6)