f[x_] := (BesselJ[0, x] + 0.5)/1.5
\[Delta]values = Table[10^x, {x, 1, Log[10, 0.05], -1./100}];
frames = Table[
   sum = Table[{\[Delta]min, Total@Table[ f[j*\[Delta]min]*\[Delta]min, {j, 1, 10/\[Delta]min}]}, {\[Delta]min, Select[\[Delta]values, # > \[Delta] &]}];
   Grid[{{
      Plot[f[x], {x, 0, 10}, PlotStyle -> {Thick, Black} , AxesStyle -> {Black, Thick}, AxesLabel -> {"x", "y"}, LabelStyle -> {Bold, Black, FontSize -> 14}, PlotRange -> {{0, 10}, {0, 1}}, Epilog -> {Orange, Opacity[0.25], EdgeForm[Directive[Thick, Orange, Opacity[0.5]]], Table[Rectangle[{j*\[Delta], 0}, {(j + 1)*\[Delta], f[j*\[Delta]]}], {j, 0, 10/\[Delta]}]}, ImageSize -> 400]
      ,
      ListLogLinearPlot[sum, Joined -> True, PlotStyle -> {Thick, Black}, Axes -> False, Frame -> True, FrameLabel -> {"Sampling interval size", "Total area of rectangles"}, LabelStyle -> {Bold, Black, FontSize -> 14}, PlotRange -> {{0.05, 10}, {0, 4.1}}, ImageSize -> 400]
      }}]
   , {\[Delta], \[Delta]values}];
ListAnimate[frames]