Riemann Sum Finder

f[x_] := Sqrt[1 - x^2];(*use this to change the function*)
Manipulate[
 intervals = N[Table[i, {i, 0, 1, 1/n}]];
 height = Map[f[#] &, intervals];
 pts = Partition[Riffle[intervals, height], 2];
 If[a == "right",
  area = height*1/n;
  areaAll = Total[area];
  tearDrier = Table[{intervals[[i]], 0}, {i, 1, Length[intervals]}];
  rect = Table[{tearDrier[[i - 1]], pts[[i]]}, {i, 1,
      Length[tearDrier]}][[2 ;; -1]]
  ,
  If[a == "left",
    area = height*1/(n - 1);
    areaAll = Total[area];
    tearDrier =
     Table[{intervals[[i]], 0}, {i, 1, Length[intervals]}][[2 ;; -1]];
    rect = Table[{tearDrier[[i]], pts[[i]]}, {i, 1, Length[tearDrier]}]
    ,
    areaAll = Total[midHeight*1/n];
    middle = (#[[1]] + #[[2]])/2 & /@ Partition[intervals, 2, 1];
    midHeight = Map[f[#] &, middle];
    midPts = Partition[Riffle[intervals[[2 ;; -1]], midHeight], 2];
    tearDrier =
     Insert[
       Table[{intervals[[i]], 0}, {i, 1, Length[intervals]}], {0, 0},
       1][[2 ;; -2]];
    rect =
     Table[{tearDrier[[i]], midPts[[i]]}, {i, 1, Length[tearDrier]}];
    ];
  ];
 Show[
  Plot[f[x], {x, 0, 1},
   PlotRange -> {0, 1},
   AspectRatio -> 1,
   PlotLabel -> Style["Area: " <> ToString[areaAll], Blue]]
  ,
  Graphics[{
    Opacity[0.25],
    EdgeForm[Blue],
    Map[Apply[Rectangle, #] &, rect]}]
  ],
 {n, 2, 200, 1}, {a, {"left", "middle", "right"}}]