#DESCRIPTION
#KEYWORDS('Integrals', 'Area and Distance')
#  Interpreting Riemann sums in terms of area
#ENDDESCRIPTION

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('Area and Distance')
## Date('5/10/2008')
## Author('Paul Pearson')
## Institution('University of Rochester')
## TitleText1('Calculus: Early Transcendentals')
## EditionText1('6')
## AuthorText1('Stewart')
## Section1('5.1')
## Problem1('2,3,4')

DOCUMENT();
loadMacros("PGbasicmacros.pl",
           "PGchoicemacros.pl",
           "PGanswermacros.pl",
           "PGgraphmacros.pl",
           "PGnumericalmacros.pl",
           "extraAnswerEvaluators.pl"
           );

TEXT(beginproblem());


# Construct a graph for the left endpoint Riemann sum,
# define the function to be graphed, and add it to the graph
$graphL = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[400,400]);
$c = random(12,17,1); # a constant for scaling the function
$f = FEQ("$c / x for x in <-1,9> using color:blue and weight:2");
$ftex = "\frac{$c}{x}";
# the parentheses around $fRefL are necessary
($fRefL) = plot_functions( $graphL, $f );


# 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.
$a = random(2,4,1); # left endpoint of interval
$b = $a+4; # right endpoint of interval
$n = 8; # number of rectangles
$deltax = ($b - $a)/$n;
foreach $k (0..$n) { @x[$k] = $a + $k * $deltax; }
foreach $k (0..$n) { @y[$k] = &{$fRefL->rule}(@x[$k]); }


# Graph the left endpoint Riemann sum
$lightblue = $graphL->im->colorAllocate(148,201,255);
$darkblue = $graphL->im->colorAllocate(100,100,255);
# Create arrays of pixel references for x and y values
foreach $k (0..8) {
   @xpixL[$k] = $graphL->ii(@x[$k]);
   @ypixL[$k] = $graphL->jj(@y[$k]);
}
$xaxisL = $graphL->jj(0);
# Plot the rectangles in the Riemann sum
foreach $k (0..$n-1) {
   $graphL->im->filledRectangle(@xpixL[$k],@ypixL[$k],@xpixL[$k+1],$xaxisL,$lightblue);
   $graphL->im->rectangle(@xpixL[$k],@ypixL[$k],@xpixL[$k+1],$xaxisL,$darkblue);
}
$graphL->lb(new Label ( 8.5,0,'x','black','right','top'));
$graphL->lb(new Label ( -0.25,8.5,'y','black','right','top'));


# Construct a graph for the right endpoint Riemann sum
$graphR = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[400,400]);
# the parentheses around $fRefR are necessary
($fRefR) = plot_functions( $graphR, $f );


# Graph the right endpoint Riemann sum
$lightblue = $graphR->im->colorAllocate(148,201,255);
$darkblue = $graphR->im->colorAllocate(100,100,255);
# Create arrays of pixel references for x and y values
foreach $k (0..8) {
   @xpixR[$k] = $graphR->ii(@x[$k]);
   @ypixR[$k] = $graphR->jj(@y[$k]);
}
$xaxisR = $graphR->jj(0);
# Plot the rectangles in the Riemann sum
foreach $k (1..$n) {
   $graphR->im->filledRectangle(@xpixR[$k-1],@ypixR[$k],@xpixR[$k],$xaxisR,$lightblue);
   $graphR->im->rectangle(@xpixR[$k-1],@ypixR[$k],@xpixR[$k],$xaxisR,$darkblue);
}
$graphR->lb(new Label ( 8.5,0,'x','black','right','top'));
$graphR->lb(new Label ( -0.25,8.5,'y','black','right','top'));



BEGIN_TEXT

The rectangles in the graph below illustrate a left endpoint Riemann sum for \( \displaystyle f(x) = $ftex \) on the interval \( \lbrack $a, $b \rbrack \).  
$BR
The value of this left endpoint Riemann sum is \{ans_rule(30)\}
$BR
$BR

$BCENTER
\{ begintable(1) \}
\{ row( image( insertGraph($graphL), height=>400, width=>400, tex_size=>800 ) ) \}
\{ row("Left endpoint Riemann sum for \( y = $ftex \) on \( \lbrack $a, $b \rbrack \)") \}
\{ endtable() \}
$ECENTER

$BR
$HR
$BR

The rectangles in the graph below illustrate a right endpoint Riemann sum for \( \displaystyle f(x) = $ftex \) on the interval \( \lbrack $a, $b \rbrack \).
$BR
The value of this right endpoint Riemann sum is \{ans_rule(30)\}
$BR
$BR

$BCENTER
\{ begintable(1) \}
\{ row( image( insertGraph($graphR), height=>400, width=>400, tex_size=>800 ) ) \}
\{ row("Right endpoint Riemann sum for \( y = $ftex \) on \( \lbrack $a, $b \rbrack \)") \}
\{ endtable() \}
$ECENTER

END_TEXT

$LeftRiemannSum = 0;
foreach $k (0..$n-1) { $LeftRiemannSum = $LeftRiemannSum + @y[$k]; }
$LeftRiemannSum = $deltax * $LeftRiemannSum;
ANS(num_cmp($LeftRiemannSum));

$RightRiemannSum = 0;
foreach $k (1..$n) { $RightRiemannSum = $RightRiemannSum + @y[$k]; }
$RightRiemannSum = $deltax * $RightRiemannSum;
ANS(num_cmp($RightRiemannSum));


SOLUTION(EV3(<<'END_SOLUTION'));
$BR
$HR
${BBOLD}Solution:${EBOLD}
$BR
$BR

(A) The left endpoint Riemann sum is 
\( f(@x[0]) \cdot 0.5 + f(@x[1]) \cdot 0.5 + \cdots + f(@x[$n-1]) \cdot 0.5
= ( @y[0] + @y[1] + \cdots + @y[7] ) \cdot 0.5 = $LeftRiemannSum.\)
$BR
$BR

(B) The right endpoint Riemann sum is 
\( f(@x[1]) \cdot 0.5 + f(@x[2]) \cdot 0.5 + \cdots + f(@x[$n]) \cdot 0.5
= ( @y[1] + @y[2] + \cdots + @y[$n] ) \cdot 0.5  = $RightRiemannSum.\)

END_SOLUTION


ENDDOCUMENT();