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

Question

I haven't been able to get this one started as well. Thanks for any help you can give me

Answer 1

A = ∫_-1¹(1 - x²)dx = [(x - x³/3) |_-1¹] = [(1 - 1/3) - (-1 +1/3)] = 2 - 2/3 = 4/3

Moments

M_x = (1/2) ∫_-1¹ [1 - x⁴] dx = (1/2)[(x - x⁵/5) |_-1¹] = (1/2) [(1 - 1/5) - (-1 + 1/5)] = (1/2)(8/5) = 4/5

M_y = ∫_-1¹ x(1 - x²)dx = [(x²/2 - x⁴/4 |_-1¹] = 0

Coordinates of Centroid

x_bar = M_y/A = 0/(4/3) = 0

y_bar = M_x/A = (4/5)/(4/3) = 0.6

Centroid (0, 0.6)

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