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Tanaquil18 merged 1 commit into from
Nov 20, 2025
Merged

4-3 Calculus Practice #15

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107 changes: 96 additions & 11 deletions source/sec-4-3-definite-integral.xml
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<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec-4-3-definite-integral">
<title>The definite integral</title>
<!-- <objectives>
<objectives>
<ul>
<li>
<!-- <li>
<p>
How does increasing the number of subintervals affect the accuracy of the approximation generated by a Riemann sum?
Understand the definite integral of a function as the limit of Riemann sum approximations as the number of rectangles gets larger and larger.
</p>
</li>

</li> -->
<li>
<p>
What is the definition of the definite integral of a function <m>f</m> over the interval <m>[a,b]</m>?
Understand the definite integral as the exact net-signed area between a function and the
<m>x-</m>axis, and be able to use known area formulas to calculate <m>\int_a^b f(x)dx</m>.
</p>
</li>

<li>
<p>
What does the definite integral measure exactly,
and what are some of the key properties of the definite integral?
Be able to use known properties of the definite integral that arise from the fact that the definite
integral measures the exact net-signed area.
</p>
</li>
</ul>
</objectives> -->
</objectives>
<subsection xml:id="subsec-definite-integral-foundations">
<title>Foundations</title>
<p>
Expand All @@ -34,10 +34,95 @@

<exercises xml:id="cp-definite-integral">
<title>Calculus Practice</title>
<introduction>
<!-- <introduction>
<p>
Coming soon.
</p>
</introduction>
</introduction> -->
<exercise label="def-int-areas-1"><title>Using areas to calculate the definite integral</title>
<webwork source="Contrib/CCCS/CalculusOne/05.2/CCD_CCCS_Openstax_Calc1_C1-2016-002_5_2_71.pg" />
</exercise>

<exercise label="def-int-areas-2"><title>Using areas to calculate the definite integral</title>
<webwork source="Contrib/CUNY/OPLremix/Library/Rochester/setIntegrals0Theory/sc5_2_24_mo.pg" />
</exercise>

<exercise label="def-int-areas-3"><title>Using areas to calculate the definite integral</title>
<webwork source="Contrib/Valdosta/APEX_Calculus/5.2/APEX_5.2_9.pg" />
</exercise>

<exercise label="def-int-areas-4"><title>Using areas to calculate the definite integral and average value</title>
<webwork source="Library/FortLewis/Calc1/05-02-Definite-integral/Definite-integral-01.pg" />
</exercise>

<exercise label="def-int-areas-5"><title>Using areas to calculate the definite integral</title>
<webwork source="Library/FortLewis/Calc1/05-02-Definite-integral/HGM5-05-02-Definite-integral-27.pg" />
</exercise>

<exercise label="def-int-areas-6"><title>Using areas to calculate the definite integral</title>
<webwork source="Library/UCSB/Stewart5_5_2/Stewart5_5_2_33/Stewart5_5_2_33.pg" />
</exercise>

<exercise label="def-int-areas-7"><title>Using areas to calculate the definite integral</title>
<webwork source="Library/UMN/calculusStewartCCC/s_5_2_31.pg" />
</exercise>

<exercise label="def-int-areas-8"><title>Using areas to calculate the definite integral</title>
<webwork source="Library/UMN/calculusTaalman/t_4_3_prob01.pg" />
</exercise>

<exercise label="def-int-areas-9"><title>Using areas to calculate the definite integral</title>
<webwork source="Library/Valdosta/APEX_Calculus/5.2/APEX_5.2_10.pg" />
</exercise>

<exercise label="def-int-properties-1"><title>Using properties of the definite integral</title>
<webwork source="Contrib/CCCS/CalculusOne/05.2/CCD_CCCS_Openstax_Calc1_C1-2016-002_5_2_88.pg" />
</exercise>

<exercise label="def-int-properties-2"><title>Using properties of the definite integral</title>
<webwork source="Library/ma122DB/set11/s5_2_37.pg" />
</exercise>

<exercise label="def-int-properties-3"><title>Using properties of the definite integral</title>
<!-- <webwork source="Library/Rochester/setIntegrals0Theory/sc5_2_30.pg" /> -->
<webwork><xi:include parse="text" href="../webworkfiles/defintproperty.pg"/></webwork>
</exercise>

<exercise label="def-int-properties-4"><title>Using properties of the definite integral</title>
<webwork source="Library/UCSB/Stewart5_5_2/Stewart5_5_2_47.pg" />
</exercise>

<exercise label="def-int-properties-5"><title>Using properties of the definite integral</title>
<webwork source="Library/UMN/calculusTaalman/t_4_3_29.pg" />
</exercise>

<exercise label="def-int-properties-6"><title>Using properties of the definite integral</title>
<webwork source="Library/Valdosta/APEX_Calculus/5.2/APEX_5.2_18-21.pg" />
</exercise>

<exercise label="def-int-average-value"><title>Approximating the average value of a continuous function</title>
<webwork source="Library/UCSB/Stewart5_6_5/Stewart5_6_5_15.pg" />
</exercise>

<exercise label="def-int-meaning-1"><title>Meaning of the definite integral</title>
<webwork source="Library/UMN/calculusStewartCCC/s_5_2_47.pg" />
</exercise>

<exercise label="def-int-meaning-2"><title>Meaning of the definite integral</title>
<webwork source="Library/UCSB/Stewart5_5_4/Stewart5_5_4_50.pg" />
</exercise>

<exercise label="def-int-meaning-3"><title>Meaning of the definite integral</title>
<webwork source="Library/FortLewis/Calc1/05-03-FTC/HGM5-05-03-FTC-13.pg" />
</exercise>

<exercise label="def-int-meaning-4"><title>Meaning of the definite integral</title>
<webwork source="Library/Michigan/Chap5Sec4/Q31.pg" />
</exercise>

<exercise label="def-int-meaning-5"><title>Meaning of the definite integral</title>
<webwork source="Contrib/NAU/setCalcI/integralAsArea.pg"/>
</exercise>

</exercises>
</section>
57 changes: 57 additions & 0 deletions webworkfiles/defintproperty.pg
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## DESCRIPTION
## Calculus
## ENDDESCRIPTION

## Tagged by tda2d

## DBsubject(Calculus - single variable)
## DBchapter(Integrals)
## DBsection(Conceptual understanding of integration)
## Institution(Rochester)
## MLT(integral_concept_partition_ab)
## MLTleader(1)
## Level(2)
## TitleText1('Calculus: Early Transcendentals')
## AuthorText1('Rogawski')
## EditionText1('1')
## Section1('5.2')
## Problem1('61')
## KEYWORDS('integral')

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

loadMacros('PGstandard.pl', 'PGML.pl', 'PGchoicemacros.pl', 'PGcourse.pl');

$showPartialCorrectAnswers = 1;

$a = random(1, 10, 1);
$add1 = random(1, 10, 1);
$add2 = random(1, 10, 1);
$b = $a + $add1;
$c = $b + $add2;

BEGIN_PGML
[``` \int_{[$a]}^{[$c]} f(x) \,dx - \int_{[$a]}^{[$b]} f(x) \,dx = \int_{a}^{b} f(x) \,dx```]


where [` a= `] [_]{$b}
and [` b= `] [_]{$c}.
END_PGML

BEGIN_PGML_SOLUTION

First recall that
[``` \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx = \int_{a}^{c} f(x) \,dx ```]
and therefore we can rearrange it to look like this:
[``` \int_{a}^{c} f(x) \,dx - \int_{a}^{b} f(x) \,dx = \int_{b}^{c} f(x) \,dx ```]


Applying that in this specific context, we find:
[```
\int_{[$a]}^{[$c]} f(x) \,dx - \int_{[$a]}^{[$b]} f(x) \,dx = \int_{[$b]}^{[$c]} f(x) \,dx
```]
Thus we find [` a=[$b]`] and [`b=[$c]`].

END_PGML_SOLUTION

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