## DESCRIPTION
## Calculus
## ENDDESCRIPTION

## KEYWORDS('integral' 'summation' 'limit')
## Tagged by tda2d

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Approximating Integrals')
## Date('')
## Author('')
## Institution('Rochester')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl",
"PGcourse.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,4,2);
$a2 = $a*2; 
$b = random(2,9,1)*random(-1,1,2);
$c = random(3,5,1);

BEGIN_TEXT

The interval \([0,$c]\) is partitioned into \(n\) equal subintervals, and a number \(x_i\) is arbitrarily chosen in the \(i^{th}\) subinterval for each \(i\). Then 
$BR 
\( \displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{$a2 x_i + $b}{n} = \) \{ans_rule(30)\} 
END_TEXT

$answer = $a*$c + $b; 

ANS(num_cmp($answer));

SOLUTION(EV3(<<'EOF'));
$SOL $BR
Let's interpret the sum as a Riemann sum. 
$BR Recall that the Riemann sum 
for a function \(f(x)\) on the interval \([0,$c]\) has the form 
\( \displaystyle \sum_{i=1}^n f(x_i) \frac{$c}{n} \) since the length of each subinterval is \( \displaystyle \Delta x = \frac{$c}{n} \). 
$BR $BR 
\( \displaystyle \sum_{i=1}^{n}\frac{$a2 x_i + $b}{n} = 
\sum_{i=1}^{n}\frac{$a2 x_i + $b}{$c} \cdot \frac{$c}{n} \), therefore the given sum is the Riemann sum for \(\displaystyle f(x) = \frac{$a2 x + $b}{$c}\). 
$BR $BR The limit of the Riemann sum as \(n\) approaches infinity is the integral of the function \(f(x)\) from \(0\) to \($c\), thus 
$BR $BR 
\( \displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{$a2 x + $b}{$c} \cdot \frac{$c}{n} = \int_0^{$c} \frac{$a2 x + $b}{$c} dx = \frac{1}{$c} \int_0^{$c} ($a2 x + $b) dx =  
\left. \frac{1}{$c} \left( $a x^2 + $b x \right) \right|_0^{$c} = 
\frac{1}{$c} \left( $a \cdot $c^2 + $b\cdot $c\right)  = $answer \)

EOF

ENDDOCUMENT();        # This should be the last executable line in the problem.