f[x_]:=x^3-3x+18
plot=Plot[f[x],{x,-3,3},PlotRange->{0,40}];
rectangles=Flatten[
Table[
Plot[f[a],{x,a,a+1},PlotStyle->Opacity[.5],Filling->Automatic,PlotRange->{0,40}],
{a,-2,3}
]
];
Show[Flatten[{plot,rectangles}]]

f[x_,y_]:=50-x^2-2y^2
plot=Plot3D[f[x,y],{x,0,4},{y,0,4}];
steps=Flatten[
Table[
Table[
Plot3D[f[a,b],{x,a,a+1},{y,b,b+1},PlotStyle->Opacity[.5],Filling->Automatic],
{a,0,3}],
{b,0,3}]
];
Show[Flatten[{plot,steps}]]

f[x_,y_]:=50-x^2-2 y^2
plot=Plot3D[f[x,y],{x,0,4},{y,0,4},PlotStyle->Black];

g[x_,y_,a_,b_]:=ConditionalExpression[f[a,b],a≤x<a+1&&b≤y<b+1]
fs[x_,y_]:=Table[g[x,y,a,b],{a,0,3},{b,0,3}]

steps=Plot3D[Evaluate[fs[x,y]],{x,0,4},{y,0,4},Filling->Automatic,PlotStyle->Directive[Opacity[.5],Red]];
Show[Flatten[{plot,steps}]]

f[x_,y_]:=16-3x^2-y^2
plot=Plot3D[f[x,y],{x,0,2},{y,0,2},PlotStyle->Black];

g[x_,y_,a_,b_]:=ConditionalExpression[f[a,b],a≤x<a+1&&b≤y<b+1]
fs[x_,y_]:=Table[g[x,y,a,b],{a,0,1},{b,0,1}]

steps=Plot3D[Evaluate[fs[x,y]],{x,0,2},{y,0,2},Filling->Automatic,PlotStyle->Directive[Opacity[.5],Red]];
Show[Flatten[{plot,steps}]]

f[x_,y_]:=16-3x^2-y^2
plot=Plot3D[f[x,y],{x,0,2},{y,0,2},PlotStyle->Black];

g[x_,y_,a_,b_]:=ConditionalExpression[f[a,b],a≤x<a+1&&b≤y<b+1]
fs[x_,y_]:=Table[g[x,y,a,b],{a,0,2},{b,0,2}]

steps=Plot3D[Evaluate[fs[x+1,y+1]],{x,0,2},{y,0,2},Filling->Automatic,PlotStyle->Directive[Opacity[.5],Red]];
Show[Flatten[{plot,steps}]]

x0=4;
y0=4;
step=.25;

f[x_,y_]:=50-x^2-2 y^2
plot=Plot3D[f[x,y],{x,0,x0},{y,0,y0},PlotStyle->Directive[Yellow,Opacity[1]]];


g[x_,y_,a_,b_]:=ConditionalExpression[f[a,b],a≤x<a+step&&b≤y<b+step]
fs[x_,y_]:=Table[g[x,y,a,b],{a,0,x0,step},{b,0,y0,step}]

steps=Plot3D[Evaluate[fs[x,y]],{x,0,x0},{y,0,y0},Filling->Automatic,PlotStyle->Directive[Opacity[.15],Red]];
Show[Flatten[{plot,steps}]]

RegionPlot[ArcTan[x,y] /(2Pi)≤ Sqrt[x^2 +y^2] ≤ArcTan[x,y]/(2Pi)+1,{x,-2,2},{y,-2,2}]

RegionPlot3D[1 ≤x^2+y^2≤4&&0 ≤ ArcTan[x,y]≤Pi/6&&1<z<2,{x,0,2},{y,-1,1},{z,0,3}]

RegionPlot3D[1 ≤x^2+y^2+z^2≤4&&0Pi/6≤ ArcTan[x,y]≤Pi/4&&Pi/6<ArcCos[z/Sqrt[x^2+y^2+z^2]]<Pi/3,{x,0,2},{y,-1,1},{z,0,3},MeshFunctions->{Sqrt[#1^2+#2^2+#3^2]&,ArcTan[#1,#2]&,ArcCos[#3/Sqrt[#1^2+#2^2+#3^2]]&}]