(* We find the centroid of the solid determined by z>=x^2+y2, z<=9, and y≥0 *)
RegionPlot3D[z≥x^2+y^2 &&z≤9&&y≥0,{x,-3,3},{y,-3,3},{z,0,9},PlotPoints->50]

x=r Cos[t]
y=r Sin[t]
Jac=Simplify[Det[D[{x,y,z},{{r,t,z}}]]]

a=9
m=Integrate[Integrate[Integrate[ 1 Jac ,{z,r^2,a}],{t,0,Pi}],{r,0,Sqrt[a]}]
mx=Integrate[Integrate[Integrate[x Jac ,{z,r^2,a}],{t,0,Pi}],{r,0,Sqrt[a]}]
my=Integrate[Integrate[Integrate[y Jac ,{z,r^2,a}],{t,0, Pi}],{r,0,Sqrt[a]}]
mz=Integrate[Integrate[Integrate[z Jac ,{z,r^2,a}],{t,0, Pi}],{r,0,Sqrt[a]}]
c={mx/m,my/m,mz/m}
c//N