x=y² and x²=8y intersect when x²=y⁴=8y, y(y³-8)=0, so y=0 and x=0, and y=2 and x=4. The integration limits will be [0,4].
The two functions are y=√x and y=x²/8.
Area A between curves=∫(√x-x²/8)dx[0,4].
A=[⅔x^(3/2)-x³/24]⁴₀=(16/3-64/24)=8/3.
Ax̄=∫x(√x-x²/8)dx[0,4]=∫(x^(3/2)-x³/8)dx[0,4].
Ax̄=[⅖x^(5/2)-x⁴/32]⁴₀=(64/5-8)=24/5.
x̄=(24/5)(3/8)=9/5.
Aȳ=½∫(x-x⁴/64)dx[0,4]=½[x²/2-x⁵/320]⁴₀=½(8-16/5)=12/5.
ȳ=(12/5)(3/8)=9/10.
Centroid is (9/5,9/10)=(1.8,0.9).