Riemann Sums

DOCUMENT();

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

install_weighted_grader();

$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

# Generate arrays of x and y values for the Riemann sum.
# There are n + 1 entries in each array so that we can use
# only one pair of arrays for both the left and the right
# endpoint Riemann sums.
$n  = 4;                # number of rectangles
$dx = ($b - $a) / $n;
for $k (0 .. $n) {
    $x[$k] = $a + $k * $dx;
    $y[$k] = $f->eval(x => $x[$k]);
}
$sumLeft  = 0;
$sumRight = 0;
for $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
);

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]`].

>>[@ image($graph1, height => 250, width => 250, tex_size => 450) @]*<<

>>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]`].

>>[@ image($graph2, height => 250, width => 250, tex_size => 450) @]*<<

>>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();