ParametricPlot[{Cos[3 t],Sin[5t]},{t,0,2Pi}]

D[{Cos[3 t],Sin[5t]},t]

{-3 Sin[3 t],5 Cos[5 t]}.{-3 Sin[3 t],5 Cos[5 t]}

Integrate[(Sqrt[25 ((Cos[5 t])^2)+9 ((Sin[3 t])^2)]),{t,0,2 Pi}]

Integrate[(Sqrt[25 ((Cos[5 t])^2)+9 ((Sin[3 t])^2)]),t]

NIntegrate[(Sqrt[25 ((Cos[5 t])^2)+9 ((Sin[3 t])^2)]),{t,0,2 Pi}]

Plot[(Sqrt[25 ((Cos[5 t])^2)+9 ((Sin[3 t])^2)]),{t,0,2 Pi},PlotRange->{0,8}]

decovase1=Show[
ParametricPlot3D[{(2+Sin[z])Cos[θ],(2+Sin[z])Sin[θ],z},{z,0,2 Pi},{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
ParametricPlot3D[{r Cos[θ], r Sin[θ],0},{r,0,2 },{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
ParametricPlot3D[{(2+Sin[(θ/8)])Cos[θ],(2+Sin[(θ/8)])Sin[θ],(θ/8)},{θ,0, 16 Pi},PlotStyle->{Thickness[0.01],Red}],ParametricPlot3D[{(2+Sin[(θ/8)])Cos[θ+(Pi/4)],(2+Sin[(θ/8)])Sin[θ+(Pi/4)],(θ/8)},{θ,0, 16 Pi},PlotStyle->{Thickness[0.01],Blue}],
ParametricPlot3D[{(2+Sin[(θ/8)])Cos[θ+(Pi/2)],(2+Sin[(θ/8)])Sin[θ+(Pi/2)],(θ/8)},{θ,0, 16 Pi},PlotStyle->{Thickness[0.01],White}],ImageSize->800]; Show[decovase1,ImageSize->400]

Show[
ParametricPlot3D[{(2+Sin[z])Cos[θ],(2+Sin[z])Sin[θ],z},{z,0,2 Pi},{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
ParametricPlot3D[{r Cos[θ], r Sin[θ],0},{r,0,2 },{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
Table[ParametricPlot3D[{(2+Sin[(θ/8)])Cos[θ+sh],(2+Sin[(θ/8)])Sin[θ+sh],(θ/8)},{θ,0, 16 Pi},PlotStyle->{Thickness[0.01],Hue[(sh/(2 Pi))]}],{sh,(Pi/4),2Pi,(Pi/4)}]]

decovase=Show[
ParametricPlot3D[{(2+Sin[z])Cos[θ],(2+Sin[z])Sin[θ],z},{z,0,2 Pi},{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
ParametricPlot3D[{r Cos[θ], r Sin[θ],0},{r,0,2 },{θ,0, 2 Pi},PlotStyle->{Opacity[0.5]},Mesh->False],
Table[ParametricPlot3D[{(2+Sin[(θ/8)])Cos[θ+sh],(2+Sin[(θ/8)])Sin[θ+sh],(θ/8)},{θ,0, 16 Pi},PlotStyle->{Thickness[0.01],Hue[(sh/(2 Pi))],Opacity[0.5]}],{sh,(Pi/4),2Pi,(Pi/4)}],ImageSize->800];
Show[decovase,ImageSize->400]

SetDirectory[NotebookDirectory[]]

Export["decovase.gif",decovase]

Export["decovase.png",decovase]

Export["decovase1.gif",decovase1]

Export["decovase1.png",decovase1]