Plane and paraboloid problem from Wednesday, March 27th

ParametricPlot3D[{{r Cos[θ], r Sin[θ], r^2},{r Cos[θ], r Sin[θ],r Sin[θ]+ 2}},{r,0,2},{θ,0,2π},PlotStyle->{{Blue,Opacity[0.5]},{Yellow,Opacity[0.5]}}]

Paraboloid and xy-plane center of mass problem from 4/1:

ParametricPlot3D[{{r Cos[θ], r Sin[θ],9- r^2},{r Cos[θ], r Sin[θ],0}},{r,0,3},{θ,0,2π},PlotStyle->{{Blue,Opacity[0.5]},{Yellow,Opacity[0.5]}}]

volume of solid using cylindrical:

vol=(∫_0^(2π))((∫_0^3)((∫_0^(9-(r^2)))1 r\[DifferentialD]z)\[DifferentialD]r)\[DifferentialD]θ

first moment using cylindrical:

Mxy=(∫_0^(2π))((∫_0^3)((∫_0^(9-(r^2)))z r\[DifferentialD]z)\[DifferentialD]r)\[DifferentialD]θ

which gives zbar, the z-coordinate of the center of mass as

zbar=(Mxy/vol)

which indeed is less than halfway up the solid, which included z-values with range 0 to 9:

ParametricPlot3D[{{r Cos[θ], r Sin[θ],9- r^2},{r Cos[θ], r Sin[θ],0},{r Cos[θ], r Sin[θ],3}},{r,0,3},{θ,0,2π},PlotStyle->{{Blue,Opacity[0.5]},{Yellow,Opacity[0.5]}}]

The second integral, to find the first moment across the xy-plane, in smaller steps:

(∫_0^(9-(r^2)))z r\[DifferentialD]z

(∫_0^3)((∫_0^(9-(r^2)))z r\[DifferentialD]z)\[DifferentialD]r