Centroid for area between y=x² and y=4. The parabola and the line intersect at x=±2. This defines the limits for integration.
Because of symmetry, we know that the centroid must be on the y-axis (x̄=0) so we only need to find ȳ. First, we need the symmetrical area A = 2∫(4-x²)dx for x∈[0,2]:
2[4x-x³/3]²₀=2(8-8/3)=32/3.
Aȳ=½∫(16-x⁴)dx[-2,2]=[8x-x⁵/10]²₋₂=(16-16/5-(-16+16/5))=128/5.
ȳ=(128/5)/(32/3)=(128/5)(3/32)=12/5=2.4.
Centroid is (0,2.4).