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}]]

f59[x_,y_]:=Piecewise[{{x*y,y<3x}}]
Plot3D[f59[x,y],{x,0,1},{y,0,3},Filling->Automatic,FillingStyle->Opacity[.7]]

f510[x_,y_]:=Piecewise[{{y Sqrt[x],1-x<y&&y<2-x}}]
Plot3D[f510[x,y],{x,0,1},{y,0,2},Filling->Automatic,FillingStyle->Opacity[.7]]

f512[x_,y_]:=Piecewise[{{10-y-x,y<10-x}}]
Plot3D[f512[x,y],{x,0,10},{y,0,10},Filling->Automatic,FillingStyle->Opacity[.7]]

f513[x_,y_]:=Piecewise[{{y(x-1),x>y&&y>x^2}}]
Plot3D[f513[x,y],{x,0,1},{y,0,1},Filling->Automatic,FillingStyle->Opacity[.7]]

f514[x_,y_]:=Piecewise[{{4-Sqrt[x^2+y^2],4-Sqrt[x^2+y^2]>0}}]
Plot3D[f514[x,y],{x,-4,4},{y,-4,4},Filling->Automatic,FillingStyle->Opacity[1]]

f515[x_,y_]:=Piecewise[{{25-x^2-y^2,x^2+y^2<16},{9,0==0}}]
Plot3D[{f515[x,y]},{x,-4,4},{y,-4,4},Filling->9,FillingStyle->Opacity[.7],PlotRange->{0,25}]

f516[x_,z_]:=Piecewise[{{Sqrt[z-x^2],x<Sqrt[z]}}]
Plot3D[f516[x,z],{z,0,3},{x,0,Sqrt[3]},Filling->Automatic,FillingStyle->Opacity[.7]]

f517[x_,y_]:=Piecewise[{{x Sqrt[y^3+1],x/3<y && y <2}}]
Plot3D[f517[x,y],{x,0,6},{y,0,2},Filling->Automatic,FillingStyle->Opacity[.7]]

f518[x_,y_]:=Piecewise[{{E^(x^2),y<x&&x<2}}]
Plot3D[f518[x,y],{x,0,2},{y,0,2},Filling->Automatic,FillingStyle->Opacity[.7]]

f519[x_,z_]:=Piecewise[{{2-x/3-2z/3,z<3-x/2}}]
Plot3D[f519[x,z],{x,0,6},{z,0,3},Filling->Automatic,FillingStyle->Opacity[.7]]

f520[x_,y_]:=Piecewise[{{x^2y,y<Sqrt[1-x^2]}}]
Plot3D[f520[x,y],{x,-1,1},{y,0,1},Filling->Automatic,FillingStyle->Opacity[.7]]

f521[x_,y_]:=Piecewise[{{x^2y^2,y<Sqrt[1-x^2]}}]
Plot3D[f520[x,y],{x,-1,1},{y,0,1},Filling->Automatic,FillingStyle->Opacity[.7]]

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]]&}]