Riemann Sums

DOCUMENT();

loadMacros(
    'PGstandard.pl', 'PGML.pl', 'parserPopUp.pl', 'PGtikz.pl',
    'PGcourse.pl'
);

$c  = random(9, 13);       # a constant for scaling the function
$f  = Compute("x^2/$c");
$a  = random(2, 5);        # left endpoint of interval
$b  = $a + 2;              # right endpoint of interval
$n  = 4;                   # number of rectangles
$dx = ($b - $a) / $n;

# Generate arrays of interval endpoints and function values.
for my $k (0 .. $n) {
    $x[$k] = $a + $k * $dx;
    $y[$k] = $f->eval(x => $x[$k]);
}

# Compute the Riemann sums.
$sumLeft  = 0;
$sumRight = 0;
for my $k (0 .. $n - 1) {
    $sumLeft  += $y[$k] * $dx;
    $sumRight += $y[ $k + 1 ] * $dx;
}

# Graph of the Left Riemann sum rectangles
$graph1 = createTikZImage();
$graph1->tikzLibraries('arrows.meta');
$graph1->BEGIN_TIKZ
\tikzset{>={Stealth[scale = 2]}}
\Large % Make the fonts a little bigger.
\filldraw[
    draw = LightBlue,
    fill = white,
    rounded corners = 10pt,
    thick,
    use as bounding box
] (-1, -1) rectangle (9, 9);
\draw[->] (-1, 0) -- (9, 0) node[above left, outer sep = 3pt] {\(x\)};
\foreach \x in {1, ..., 8} \draw (\x, 5pt) -- (\x, -5pt) node [below] {\(\x\)};
\draw[->] (0, -1) -- (0, 9) node[below right, outer sep = 3pt] {\(y\)};
\foreach \y in {1, ..., 8} \draw (5pt, \y) -- (-5pt, \y) node[left] {\(\y\)};
\draw[<->] plot[domain = -1:9] (\x, {\x * \x / $c});
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[0], 0) rectangle ($x[1], $y[0]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[1], 0) rectangle ($x[2], $y[1]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[2], 0) rectangle ($x[3], $y[2]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[3], 0) rectangle ($x[4], $y[3]);
END_TIKZ

# Graph of the right Riemann sum rectangles
$graph2 = createTikZImage();
$graph2->tikzLibraries('arrows.meta');
$graph2->BEGIN_TIKZ
\tikzset{>={Stealth[scale = 2]}}
\Large % Make the fonts a little bigger.
\filldraw[
    draw = LightBlue,
    fill = white,
    rounded corners = 10pt,
    thick,
    use as bounding box
] (-1, -1) rectangle (9, 9);
\draw[->] (-1, 0) -- (9, 0) node[above left, outer sep = 3pt] {\(x\)};
\foreach \x in {1, ..., 8} \draw (\x, 5pt) -- (\x, -5pt) node [below] {\(\x\)};
\draw[->] (0, -1) -- (0, 9) node[below right, outer sep = 3pt] {\(y\)};
\foreach \y in {1,...,8} \draw (5pt, \y) -- (-5pt, \y) node[left] {\(\y\)};
\draw[<->] plot[domain = -1:9] (\x, {\x * \x / $c});
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[0], 0) rectangle ($x[1], $y[1]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[1], 0) rectangle ($x[2], $y[2]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[2], 0) rectangle ($x[3], $y[3]);
\filldraw[draw = blue, fill = blue!50!white, fill opacity = 0.5]
    ($x[3], 0) rectangle ($x[4], $y[4]);
END_TIKZ

$leftEstimateDropDown = DropDown(
    [
        'an overestimate of',
        'equal to',
        'an underestimate of',
        'there is ambiguity'
    ],
    2
);

$rightEstimateDropDown = DropDown(
    [
        'an overestimate of',
        'equal to',
        'an underestimate of',
        'there is ambiguity'
    ],
    0
);

$leftRiemannSumAltText =
    "The graph of a parabola opening up with vertex at the point (0, 0) is "
    . "shown. Rectangles above the intervals from $x[0] to $x[1], $x[1] to "
    . "$x[2], $x[2] to $x[3], and $x[3] to $x[4] with heights $y[0], $y[1], "
    . "$y[2], and $y[3], respectively are also shown.";

$rightRiemannSumAltText =
    "The graph of a parabola opening up with vertex at the point (0, 0) is "
    . "shown. Rectangles above the intervals from $x[0] to $x[1], $x[1] to "
    . "$x[2], $x[2] to $x[3], and $x[3] to $x[4] with heights $y[1], $y[2], "
    . "$y[3], and $y[4], respectively are also shown.";

BEGIN_PGML
Suppose [``f(x) = \frac{x^2}{[$c]}``].

a. The rectangles in the graph below illustrate a left endpoint Riemann sum for
[`f(x)`] on the interval [`[$a] \leq x \leq [$b]`].  The value of this left
endpoint Riemann sum is [_]{Real($sumLeft)->cmp(weight => 45)}{5} and it is
[_]{ $leftEstimateDropDown->cmp(weight => 5) } the area of the region enclosed
by [`y = f(x)`], the [`x`]-axis, and the vertical lines [`x = [$a]`] and
[`x = [$b]`].

>>[![$leftRiemannSumAltText]!]{$graph1}{250}<<  
>>Left endpoint Riemann sum<<

b. The rectangles in the graph below illustrate a right endpoint Riemann sum for
[`f(x)`] on the interval [`[$a] \leq x \leq [$b]`].  The value of this right
endpoint Riemann sum is [_]{ Real($sumRight)->cmp(weight => 45) }{5} and it is
[_]{ $rightEstimateDropDown->cmp(weight => 5) } the area of the region enclosed
by [`y = f(x)`], the [`x`]-axis, and the vertical lines [`x = [$a]`] and
[`x = [$b]`].

>>[![$rightRiemannSumAltText]!]{$graph2}{250}<<  
>>Right endpoint Riemann sum<<
END_PGML

BEGIN_PGML_SOLUTION
a. The left endpoint Riemann sum is
[```
    f([$x[0]]) \cdot 0.5 + f([$x[1]]) \cdot 0.5 + \cdots + f([$x[ $n - 1]]) \cdot 0.5
    = ([$y[0]] + [$y[1]] + \cdots + [$y[ $n-1 ]]) \cdot 0.5 = [$sumLeft].
```]

b. The right endpoint Riemann sum is
[```
    f([$x[1]]) \cdot 0.5 + f([$x[2]]) \cdot 0.5 + \cdots + f([$x[ $n ]]) \cdot 0.5
    = ([$y[1]] + [$y[2]] + \cdots + [$y[ $n ]]) \cdot 0.5  = [$sumRight].
```]
END_PGML_SOLUTION

ENDDOCUMENT();