A problem with Manipulate

Question

I have:

Clear[f, a, b, n, dx];
f[x_] = x^2;
a = 0; b = 1; dx = (b - a)/n;
Manipulate[
 Show[Plot[f[x], {x, a, b},
   PlotStyle -> Thick,
   AxesLabel -> {x, y}],
  Graphics[{
    Table[
     {Opacity[0.05], EdgeForm[Gray],
      Rectangle[{a + i dx, 0}, {a + (i + 1) dx, f[a + i dx]}]}, {i, 0,
       n - 1, 1}]
    }]],
 {{n, 10}, 10, 50, 10}
 ]

But I get an error:

Coordinate {$CellContext`n^(-1), 0} should be a pair of numbers, or a Scaled or Offset form.

Can someone tell me what I am doing wrong?

Thanks to Mahdi, I now have:

Manipulate[Module[{a = 0, b = 1, dx, f, rightSum},
  dx = (b - a)/n;
  f[x_] = x^2;
  rightSum = N@Sum[f[a + i dx] dx, {i, 1, n}];
  Show[Plot[f[x], {x, a, b},
    PlotStyle -> Thick,
    AxesLabel -> {"x", "y"}],
   Graphics[{
     Table[{Opacity[0.05], EdgeForm[Gray], 
       Rectangle[{a + i dx, 0}, {a + (i + 1) dx, 
         f[a + (i + 1) dx]}]}, {i, 0, n - 1, 1}],
     Text[
      "N = " <> ToString[n] <> ",    R = " <> 
       ToString[rightSum], {(a + b)/2, f[b]}]
     }]]],
 {{n, 10}, 10, 50, 10, Appearance -> "Labeled"}]

Which produces this image:

Couple things I learned:

1. I tried:

   Module[{a = 0, b = 1, dx=(b-a)/n, f, rightSum}

It worked, but when I changed the b=1 to a b=2, it did not work. So, apparently the dx=(b-a)/n does not use the b=1 in the module.

2. On the other hand, when I put it inside the module,

   Manipulate[Module[{a = 0, b = 1, dx, f, leftSum}, dx = (b - a)/n;

it worked.

3. Thus far, I have two manipulates in my notebook, which don't seem to interfere with one another or all of the static code I have in the notebook.

Difficulty with Initialization: After reading Martin John Hadley's comment, I tried the following:

Manipulate[
 Show[Plot[f[x], {x, a, b},
   PlotStyle -> Thick,
   AxesLabel -> {"x", "y"}],
  Graphics[{
    Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + i dx[n]]}]}, {i, 0, n - 1}],
    Text["N = " <> ToString[n] <> ",    L = " <> 
      ToString[leftSum], {(a + b)/2, f[b]}]
    }]],
 {{n, 10}, 10, 50, 10, Appearance -> "Labeled"},
 Initialization :> (
   dx[n_] := (b - a)/n;
   f[x_] = x^2;
   leftSum = N@Sum[f[a + i dx[n]] dx[n], {i, 0, n - 1}];
   a = 0;
   b = 2
   )]

But it didn't work. I got errors such as:

Coordinate {Rational[1, 5][10], 0} should be a pair of numbers, or a Scaled or Offset form.

I cannot determine what is wrong here? Any thoughts?

Answer to my Own Question: I should have put a=0 and b=2 as the first two lines in the initialization block.

Final Result due to help from Martin John Hadley:

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}], 
    Text["N = " <> ToString[n] <> ",    R = " <> 
      ToString[rightSum[n]], {(a + b)/2, f[b]}]}]], {{n, 10}, 10, 50, 
  10, Appearance -> "Labeled"},
 Initialization :> (a = 0; b = 1; dx[n_] := (b - a)/n; f[x_] := x^2;
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

And the output image:

Thanks to both Martin and Mahdi for tremendous help.

Answer 1

As mentioned in my comment on Mahdi's answer, it is generally not advisable to use Module within Manipulate - https://mathematica.stackexchange.com/a/80324/1952 gives a good explanation of why.

I have refactored your code into a Manipulate that uses Initialization, as advised by Mahdi it is necessary to change your definition of f to use SetDelayed rather than Set and I also created a function dx dependent on the variable n:

Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}]}]],
 {{n, 10}, 10, 50, 10, Appearance -> "Labeled"},
 Initialization :> (
   a = 0; b = 1; dx[n_] := (b - a)/n; f[x_] := x^2; 
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}]
   )
 ]

Edit

My initial DynamicModule solution did not work in a clean kernel due to the same issue I helped someone with recently, content in Initialization is not evaluated until after the construct is first displayed - https://mathematica.stackexchange.com/a/80767/1952.

I've modified the DynamicModule to assign a and b within the variable specification and f within the body of the DynamicModule

 DynamicModule[{n, a = 0, b = 1, f},
 f[x_] := x^2;
 Panel[Column[
   {
    Control[{n, 10, 50, 10}], 
    Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
      AxesLabel -> {"x", "y"}, ImageSize -> 500], 
     Graphics[{Dynamic@
        Table[{Opacity[0.05], EdgeForm[Gray], 
          Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
            f[a + (i + 1) dx[n]]}]}, {i, 0, n - 1, 1}]}]]
    }
   ]
  ], Initialization :> (dx[n_] := (b - a)/n; 
   rightSum[n_] := N@Sum[f[a + i dx[n]] dx[n], {i, 1, n}])]

Answer 2

You need to specify dx as a function of n, or replace dx definition directly in Manipulate.

Functional Form of dx

Clear[f, a, b, n, dx];
f[x_] := x^2;
a = 0; b = 1;
dx[n_] := (b - a)/n;
Manipulate[
 Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
   AxesLabel -> {"x", "y"}], 
  Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
      Rectangle[{a + i dx[n], 0}, {a + (i + 1) dx[n], 
        f[a + i dx[n]]}]}, {i, 0, n - 1, 1}]}]
  ], {{n, 10}, 10, 50, 10}]

Local variables within Manipulate

To have a,b and f as local variables, you could use Module:

Manipulate[
 Module[{a = 0, b = 1, f},
  f[x_] := x^2;
  Show[Plot[f[x], {x, a, b}, PlotStyle -> Thick, 
    AxesLabel -> {"x", "y"}], 
   Graphics[{Table[{Opacity[0.05], EdgeForm[Gray], 
       Rectangle[{a + i (b - a)/n, 0}, {a + (i + 1) (b - a)/n, 
         f[a + i (b - a)/n]}]}, {i, 0, n - 1, 1}]}]]
  ], {{n, 10}, 10, 50, 10}] 

Both methods result in the following: