Neb: what to do about stale content

neb-stale-content.md

1. Downloaded example content from Thomas ~/Downloads/new\ stale\ backup.zip.

2. Unzip the original content col11963_1.5.2_from_batch and modified content col11963_1.5.2_modified_by_me:

   unzip -x ~/Downloads/new\ stale\ backup.zip col11963_1.5.2_modified_by_me/*
   
   unzip -x ~/Downloads/new\ stale\ backup.zip col11963_1.5.2_from_batch/*
   

3. Store the original content as index.cnxml.bak in the same directory as the modified content:

   find col11963_1.5.2_from_batch -name index.cnxml | while read f; do cp "$f" "${f/from_batch/modified_by_me}.bak"; done
   

4. Delete the directory with the original content:

   rm -rf col11963_1.5.2_from_batch
   

5. Grab a newer version of the content using neb:

   neb get -t prod col11963 1.6.1
   

6. Copy the index.cnxml to index.cnxml.bak so we can store the original:

   find col11963_1.6.1/ -name index.cnxml | while read f; do cp "$f" "$f.bak"; done
   

7. Do a 3-way merge of the content modified by the user and the new version 1.6.1 using diff3:

   find col11963_1.5.2_modified_by_me/ -name index.cnxml | while read f; do diff3 --merge "$f" "$f.bak" "${f/1.5.2_modified_by_me/1.6.1}.bak" >"${f/1.5.2_modified_by_me/1.6.1}"; done
   

   Note: there was a file not found error (probably due to unicode file path) so one file couldn't be merged -_-

8. Check the diff from 1.5.2 and 1.6.1 to make sure the changes in 1.6.1 looks sane:

   find col11963_1.5.2_modified_by_me/ -name index.cnxml | sort | while read f; do diff --color=always -u "$f.bak" "$f"; done | less -R
   find col11963_1.6.1/ -name index.cnxml | sort | while read f; do diff --color=always -u "$f.bak" "$f"; done | less -R
   

---

This is the diff from 1.5.2:

--- "col11963_1.5.2_modified_by_me/Calculus/05 Integration/02 The Definite Integral/index.cnxml.bak"	2019-10-04 13:59:43.567758229 +0100
+++ "col11963_1.5.2_modified_by_me/Calculus/05 Integration/02 The Definite Integral/index.cnxml"	2019-09-30 16:59:02.000000000 +0100
@@ -746,6 +746,10 @@
 <para id="fs-id1170572351512">The integrand is odd; the integral is zero.</para>
 </solution>
 </exercise>
+<para id="fs-id11705723515221">
+In the following exercises, find the net signed area between <m:math><m:mi>f</m:mi><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:math> and the x-axis.
+</para>
+
 <exercise id="fs-id1170572351517">
 <problem id="fs-id1170572351520">
 <para id="fs-id1170572351522"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:msubsup><m:mo>∫</m:mo><m:mn>1</m:mn><m:mn>3</m:mn></m:msubsup><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>2</m:mn><m:mo>−</m:mo><m:mi>x</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math> (<emphasis effect="italics">Hint:</emphasis> Look at the graph of <emphasis effect="italics">f</emphasis>.)</para>
@@ -1110,4 +1114,4 @@
 <meaning id="fs-id1170571613101">indicates which variable you are integrating with respect to; if it is <emphasis effect="italics">x</emphasis>, then the function in the integrand is followed by <emphasis effect="italics">dx</emphasis></meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.5.2_modified_by_me/Calculus/07 Techniques of Integration/03 Trigonometric Substitution/index.cnxml.bak"	2019-10-04 13:59:43.547758213 +0100
+++ "col11963_1.5.2_modified_by_me/Calculus/07 Techniques of Integration/03 Trigonometric Substitution/index.cnxml"	2019-09-30 16:53:50.000000000 +0100
@@ -433,7 +433,7 @@
 </exercise>
 <exercise id="fs-id1165040775694">
 <problem id="fs-id1165040775696">
-<para id="fs-id1165040775698"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mfrac><m:mrow><m:msup><m:mi>θ</m:mi><m:mn>3</m:mn></m:msup><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow><m:mrow><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:mfrac><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></para>
+<para id="fs-id1165040775698"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mfrac><m:mrow><m:msup><m:mi>θ</m:mi><m:mn>3</m:mn></m:msup><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow><m:mrow><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:mfrac></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></para>
 </problem>
 <solution id="fs-id1165041957568">
 <para id="fs-id1165041957570"><m:math><m:mrow><m:mo>−</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>18</m:mn><m:mo>+</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:math></para>
@@ -697,4 +697,4 @@
 <meaning id="fs-id1165041956792">an integration technique that converts an algebraic integral containing expressions of the form <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mo>,</m:mo></m:mrow></m:math> <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mo>,</m:mo></m:mrow></m:math> or <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:math> into a trigonometric integral</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.5.2_modified_by_me/Calculus/08 Introduction to Differential Equations/02 Direction Fields and Numerical Methods/index.cnxml.bak"	2019-10-04 13:59:43.623758274 +0100
+++ "col11963_1.5.2_modified_by_me/Calculus/08 Introduction to Differential Equations/02 Direction Fields and Numerical Methods/index.cnxml"	2019-09-30 16:46:05.000000000 +0100
@@ -135,7 +135,7 @@
 <para id="fs-id1170573414525">Go to this <link url="http://www.openstax.org/l/20_DifferEq">Java applet</link> and this <link url="http://www.openstax.org/l/20_SlopeFields">website</link> to see more about slope fields.</para>
 </note>
 <para id="fs-id1170570993737">Now consider the direction field for the differential equation <m:math><m:mrow><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo><m:mo>,</m:mo></m:mrow></m:math> shown in <link target-id="CNX_Calc_Figure_08_02_005"/>. This direction field has several interesting properties. First of all, at <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> and <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>2</m:mn><m:mo>,</m:mo></m:mrow></m:math> horizontal dashes appear all the way across the graph. This means that if <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn><m:mo>,</m:mo></m:mrow></m:math> then <m:math><m:mrow><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Substituting this expression into the right-hand side of the differential equation gives</para>
-<equation id="fs-id1170570995991" class="unnumbered"><label/><m:math><m:mtable><m:mtr><m:mtd columnalign="left"><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mo stretchy="false">(</m:mo><m:mn>−2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mn>0</m:mn></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>.</m:mo></m:mtd></m:mtr></m:mtable></m:math></equation>
+<equation id="fs-id1170570995991" class="unnumbered"><label/><m:math><m:mtable><m:mtr><m:mtd columnalign="left"><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mo stretchy="false">(</m:mo><m:mn>-2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>−2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mn>0</m:mn></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>.</m:mo></m:mtd></m:mtr></m:mtable></m:math></equation>
 <para id="fs-id1170571260861">Therefore <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> is a solution to the differential equation. Similarly, <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>2</m:mn></m:mrow></m:math> is a solution to the differential equation. These are the only constant-valued solutions to the differential equation, as we can see from the following argument. Suppose <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mi>k</m:mi></m:mrow></m:math> is a constant solution to the differential equation. Then <m:math><m:mrow><m:msup><m:mi>y</m:mi><m:mo>′</m:mo></m:msup><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Substituting this expression into the differential equation yields <m:math><m:mrow><m:mn>0</m:mn><m:mo>=</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:msup><m:mi>k</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math> This equation must be true for all values of <m:math><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo></m:mrow></m:math> so the second factor must equal zero. This result yields the equation <m:math><m:mrow><m:msup><m:mi>k</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> The solutions to this equation are <m:math><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> and <m:math><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>2</m:mn><m:mo>,</m:mo></m:mrow></m:math> which are the constant solutions already mentioned. These are called the equilibrium solutions to the differential equation.</para>
 <figure id="CNX_Calc_Figure_08_02_005">
 <media id="fs-id1170573263610" alt="A direction field for the given differential equation. The arrows are horizontal and pointing to the right at y = -4, y = 4, and x = 6. The closer the arrows are to x = 6, the more horizontal the arrows become. The further away, the more vertical they are. The arrows point down for y &gt; 4 and x &lt; 4, -4 &lt; y &lt; 4 and x &gt; 6, and y &lt; -4 and x &lt; 6. In all other areas, the arrows are pointing up.">
@@ -1002,4 +1002,4 @@
 <meaning id="fs-id1170571042413">the increment <m:math><m:mi>h</m:mi></m:math> that is added to the <m:math><m:mi>x</m:mi></m:math> value at each step in Euler’s Method</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.5.2_modified_by_me/Calculus/09 Sequences and Series/01 Sequences/index.cnxml.bak"	2019-10-04 13:59:43.715758347 +0100
+++ "col11963_1.5.2_modified_by_me/Calculus/09 Sequences and Series/01 Sequences/index.cnxml"	2019-09-30 17:05:42.000000000 +0100
@@ -260,7 +260,9 @@
 <section id="fs-id1169736843616">
 <title>Proof</title>
 <para id="fs-id1169736843622">We prove part iii.</para>
-<para id="fs-id1169736843625">Let <m:math><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>A</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant positive integer <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> such that for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>B</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:math> such that <m:math><m:mrow><m:mo>|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mi>ε</m:mi><m:mtext>/</m:mtext><m:mn>2</m:mn></m:mrow></m:math> for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Let <m:math><m:mi>N</m:mi></m:math> be the largest of <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> and <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Therefore, for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mi>N</m:mi><m:mo>,</m:mo></m:mrow></m:math></para>
+<para id="fs-id1169736843625">Let <m:math><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>A</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant positive integer <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> such that
+<m:math><m:mo>|</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>-</m:mo><m:mi>A</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac></m:math>
+ for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>B</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:math> such that <m:math><m:mrow><m:mo>|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mi>ε</m:mi><m:mtext>/</m:mtext><m:mn>2</m:mn></m:mrow></m:math> for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Let <m:math><m:mi>N</m:mi></m:math> be the larger of <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> and <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Therefore, for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mi>N</m:mi><m:mo>,</m:mo></m:mrow></m:math></para>
 <para id="fs-id1169736852772"><m:math><m:mrow><m:mo stretchy="false">|</m:mo><m:mo stretchy="false">(</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>+</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo stretchy="false">)</m:mo><m:mtext>−</m:mtext><m:mo stretchy="false">(</m:mo><m:mi>A</m:mi><m:mo>+</m:mo><m:mi>B</m:mi><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">|</m:mo><m:mo>≤</m:mo><m:mo stretchy="false">|</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>A</m:mi><m:mo stretchy="false">|</m:mo><m:mo>+</m:mo><m:mo stretchy="false">|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo stretchy="false">|</m:mo><m:mo>&lt;</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac><m:mo>+</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac><m:mo>=</m:mo><m:mi>ε</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
 <para id="fs-id1169739110576">□</para>
 <para id="fs-id1169739110579">The algebraic limit laws allow us to evaluate limits for many sequences. For example, consider the sequence <m:math><m:mrow><m:mrow><m:mo>{</m:mo><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:msup><m:mi>n</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:mfrac></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math> As shown earlier, <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:mn>1</m:mn><m:mtext>/</m:mtext><m:mi>n</m:mi><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Similarly, for any positive integer <m:math><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo></m:mrow></m:math> we can conclude that</para>
@@ -1128,4 +1130,4 @@
 <meaning id="fs-id1169736634914">a sequence that is not bounded is called unbounded</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.5.2_modified_by_me/Calculus/15 Multiple Integration/04 Triple Integrals/index.cnxml.bak"	2019-10-04 13:59:43.671758312 +0100
+++ "col11963_1.5.2_modified_by_me/Calculus/15 Multiple Integration/04 Triple Integrals/index.cnxml"	2019-09-30 16:50:24.000000000 +0100
@@ -231,12 +231,12 @@
 <exercise xmlns:data="http://www.w3.org/TR/html5/dom.html#custom-data-attribute" id="fs-id1167794296744"><problem id="fs-id1167794296746">
 <title>Changing the Order of Integration</title>
 <para id="fs-id1167794296751">Consider the iterated integral</para>
-<equation id="fs-id1167794296754" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:mi>y</m:mi><m:mrow/></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
+<equation id="fs-id1167794296754" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mrow/></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793544547">The order of integration here is first with respect to <emphasis effect="italics">z</emphasis>, then <emphasis effect="italics">y</emphasis>, and then <emphasis effect="italics">x</emphasis>. Express this integral by changing the order of integration to be first with respect to <emphasis effect="italics">x</emphasis>, then <emphasis effect="italics">z</emphasis>, and then <m:math><m:mrow><m:mi>y</m:mi><m:mo>.</m:mo></m:mrow></m:math> Verify that the value of the integral is the same if we let <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>x</m:mi><m:mi>y</m:mi><m:mi>z</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
 </problem>
 <solution id="fs-id1167793612493">
 <para id="fs-id1167793612495">The best way to do this is to sketch the region <m:math><m:mi>E</m:mi></m:math> and its projections onto each of the three coordinate planes. Thus, let</para>
-<equation id="fs-id1167793612502" class="unnumbered"><label/><m:math><m:mrow><m:mi>E</m:mi><m:mo>=</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="false">)</m:mo><m:mo>|</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>x</m:mi><m:mo>≤</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>y</m:mi><m:mo>≤</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>z</m:mi><m:mo>≤</m:mo><m:mi>y</m:mi></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
+<equation id="fs-id1167793612502" class="unnumbered"><label/><m:math><m:mrow><m:mi>E</m:mi><m:mo>=</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="false">)</m:mo><m:mo>|</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>x</m:mi><m:mo>≤</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>y</m:mi><m:mo>≤</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>z</m:mi><m:mo>≤</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793432448">and</para>
 <equation id="fs-id1167793432451" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>=</m:mo><m:mstyle displaystyle="true"><m:mrow><m:munder><m:mo>∭</m:mo><m:mi>E</m:mi></m:munder><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>V</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793424343">We need to express this triple integral as</para>
@@ -796,4 +796,4 @@
 <meaning id="fs-id1167793512095">the triple integral of a continuous function <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> over a rectangular solid box <m:math><m:mi>B</m:mi></m:math> is the limit of a Riemann sum for a function of three variables, if this limit exists</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>

This is the diff from 1.6.1:

--- "col11963_1.6.1/Calculus/05 Integration/02 The Definite Integral/index.cnxml.bak"	2019-10-04 14:28:47.107009418 +0100
+++ "col11963_1.6.1/Calculus/05 Integration/02 The Definite Integral/index.cnxml"	2019-10-04 14:38:19.770600610 +0100
@@ -746,6 +746,10 @@
 <para id="fs-id1170572351512">The integrand is odd; the integral is zero.</para>
 </solution>
 </exercise>
+<para id="fs-id11705723515221">
+In the following exercises, find the net signed area between <m:math><m:mi>f</m:mi><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:math> and the x-axis.
+</para>
+
 <exercise id="fs-id1170572351517">
 <problem id="fs-id1170572351520">
 <para id="fs-id1170572351522"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:msubsup><m:mo>∫</m:mo><m:mn>1</m:mn><m:mn>3</m:mn></m:msubsup><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>2</m:mn><m:mo>−</m:mo><m:mi>x</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math> (<emphasis effect="italics">Hint:</emphasis> Look at the graph of <emphasis effect="italics">f</emphasis>.)</para>
@@ -1110,4 +1114,4 @@
 <meaning id="fs-id1170571613101">indicates which variable you are integrating with respect to; if it is <emphasis effect="italics">x</emphasis>, then the function in the integrand is followed by <emphasis effect="italics">dx</emphasis></meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.6.1/Calculus/07 Techniques of Integration/03 Trigonometric Substitution/index.cnxml.bak"	2019-10-04 14:28:47.083009439 +0100
+++ "col11963_1.6.1/Calculus/07 Techniques of Integration/03 Trigonometric Substitution/index.cnxml"	2019-10-04 14:38:19.678600672 +0100
@@ -431,7 +431,7 @@
 </exercise>
 <exercise id="fs-id1165040775694">
 <problem id="fs-id1165040775696">
-<para id="fs-id1165040775698"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mfrac><m:mrow><m:msup><m:mi>θ</m:mi><m:mn>3</m:mn></m:msup><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow><m:mrow><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:mfrac><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></para>
+<para id="fs-id1165040775698"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mfrac><m:mrow><m:msup><m:mi>θ</m:mi><m:mn>3</m:mn></m:msup><m:mi>d</m:mi><m:mi>θ</m:mi></m:mrow><m:mrow><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:mfrac></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></para>
 </problem>
 <solution id="fs-id1165041957568">
 <para id="fs-id1165041957570"><m:math><m:mrow><m:mo>−</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac><m:msqrt><m:mrow><m:mn>9</m:mn><m:mo>−</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>18</m:mn><m:mo>+</m:mo><m:msup><m:mi>θ</m:mi><m:mn>2</m:mn></m:msup></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:math></para>
@@ -695,4 +695,4 @@
 <meaning id="fs-id1165041956792">an integration technique that converts an algebraic integral containing expressions of the form <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mo>,</m:mo></m:mrow></m:math> <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt><m:mo>,</m:mo></m:mrow></m:math> or <m:math><m:mrow><m:msqrt><m:mrow><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:msup><m:mi>a</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:math> into a trigonometric integral</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.6.1/Calculus/08 Introduction to Differential Equations/02 Direction Fields and Numerical Methods/index.cnxml.bak"	2019-10-04 14:28:47.159009374 +0100
+++ "col11963_1.6.1/Calculus/08 Introduction to Differential Equations/02 Direction Fields and Numerical Methods/index.cnxml"	2019-10-04 14:38:20.002600453 +0100
@@ -135,7 +135,7 @@
 <para id="fs-id1170573414525">Go to this <link url="http://www.openstax.org/l/20_DifferEq">Java applet</link> and this <link url="http://www.openstax.org/l/20_SlopeFields">website</link> to see more about slope fields.</para>
 </note>
 <para id="fs-id1170570993737">Now consider the direction field for the differential equation <m:math><m:mrow><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo><m:mo>,</m:mo></m:mrow></m:math> shown in <link target-id="CNX_Calc_Figure_08_02_005"/>. This direction field has several interesting properties. First of all, at <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> and <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>2</m:mn><m:mo>,</m:mo></m:mrow></m:math> horizontal dashes appear all the way across the graph. This means that if <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn><m:mo>,</m:mo></m:mrow></m:math> then <m:math><m:mrow><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Substituting this expression into the right-hand side of the differential equation gives</para>
-<equation id="fs-id1170570995991" class="unnumbered"><label/><m:math><m:mtable><m:mtr><m:mtd columnalign="left"><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mo stretchy="false">(</m:mo><m:mn>−2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mn>0</m:mn></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>.</m:mo></m:mtd></m:mtr></m:mtable></m:math></equation>
+<equation id="fs-id1170570995991" class="unnumbered"><label/><m:math><m:mtable><m:mtr><m:mtd columnalign="left"><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:msup><m:mo stretchy="false">(</m:mo><m:mn>-2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>−2</m:mn><m:mo stretchy="false">)</m:mo><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo stretchy="false">)</m:mo></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mn>0</m:mn></m:mtd></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="left"><m:mo>=</m:mo><m:mi>y</m:mi><m:mo>′</m:mo><m:mo>.</m:mo></m:mtd></m:mtr></m:mtable></m:math></equation>
 <para id="fs-id1170571260861">Therefore <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> is a solution to the differential equation. Similarly, <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>2</m:mn></m:mrow></m:math> is a solution to the differential equation. These are the only constant-valued solutions to the differential equation, as we can see from the following argument. Suppose <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mi>k</m:mi></m:mrow></m:math> is a constant solution to the differential equation. Then <m:math><m:mrow><m:msup><m:mi>y</m:mi><m:mo>′</m:mo></m:msup><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Substituting this expression into the differential equation yields <m:math><m:mrow><m:mn>0</m:mn><m:mo>=</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>−</m:mo><m:mn>3</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:msup><m:mi>k</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math> This equation must be true for all values of <m:math><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo></m:mrow></m:math> so the second factor must equal zero. This result yields the equation <m:math><m:mrow><m:msup><m:mi>k</m:mi><m:mn>2</m:mn></m:msup><m:mo>−</m:mo><m:mn>4</m:mn><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> The solutions to this equation are <m:math><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>−2</m:mn></m:mrow></m:math> and <m:math><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>2</m:mn><m:mo>,</m:mo></m:mrow></m:math> which are the constant solutions already mentioned. These are called the equilibrium solutions to the differential equation.</para>
 <figure id="CNX_Calc_Figure_08_02_005">
 <media id="fs-id1170573263610" alt="A direction field for the given differential equation. The arrows are horizontal and pointing to the right at y = -4, y = 4, and x = 6. The closer the arrows are to x = 6, the more horizontal the arrows become. The further away, the more vertical they are. The arrows point down for y &gt; 4 and x &lt; 4, -4 &lt; y &lt; 4 and x &gt; 6, and y &lt; -4 and x &lt; 6. In all other areas, the arrows are pointing up.">
@@ -1002,4 +1002,4 @@
 <meaning id="fs-id1170571042413">the increment <m:math><m:mi>h</m:mi></m:math> that is added to the <m:math><m:mi>x</m:mi></m:math> value at each step in Euler’s Method</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.6.1/Calculus/09 Sequences and Series/01 Sequences/index.cnxml.bak"	2019-10-04 14:28:47.251009294 +0100
+++ "col11963_1.6.1/Calculus/09 Sequences and Series/01 Sequences/index.cnxml"	2019-10-04 14:38:20.434600163 +0100
@@ -260,7 +260,9 @@
 <section id="fs-id1169736843616">
 <title>Proof</title>
 <para id="fs-id1169736843622">We prove part iii.</para>
-<para id="fs-id1169736843625">Let <m:math><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>A</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant positive integer <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> such that for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>B</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:math> such that <m:math><m:mrow><m:mo>|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mi>ε</m:mi><m:mtext>/</m:mtext><m:mn>2</m:mn></m:mrow></m:math> for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Let <m:math><m:mi>N</m:mi></m:math> be the largest of <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> and <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Therefore, for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mi>N</m:mi><m:mo>,</m:mo></m:mrow></m:math></para>
+<para id="fs-id1169736843625">Let <m:math><m:mrow><m:mi>ϵ</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>A</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant positive integer <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> such that
+<m:math><m:mo>|</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>-</m:mo><m:mi>A</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac></m:math>
+ for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Since <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>=</m:mo><m:mi>B</m:mi><m:mo>,</m:mo></m:mrow></m:math> there exists a constant <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub></m:mrow></m:math> such that <m:math><m:mrow><m:mo>|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo>|</m:mo><m:mo>&lt;</m:mo><m:mi>ε</m:mi><m:mtext>/</m:mtext><m:mn>2</m:mn></m:mrow></m:math> for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Let <m:math><m:mi>N</m:mi></m:math> be the larger of <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>1</m:mn></m:msub></m:mrow></m:math> and <m:math><m:mrow><m:msub><m:mi>N</m:mi><m:mn>2</m:mn></m:msub><m:mo>.</m:mo></m:mrow></m:math> Therefore, for all <m:math><m:mrow><m:mi>n</m:mi><m:mo>≥</m:mo><m:mi>N</m:mi><m:mo>,</m:mo></m:mrow></m:math></para>
 <para id="fs-id1169736852772"><m:math><m:mrow><m:mo stretchy="false">|</m:mo><m:mo stretchy="false">(</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>+</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo stretchy="false">)</m:mo><m:mtext>−</m:mtext><m:mo stretchy="false">(</m:mo><m:mi>A</m:mi><m:mo>+</m:mo><m:mi>B</m:mi><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">|</m:mo><m:mo>≤</m:mo><m:mo stretchy="false">|</m:mo><m:msub><m:mi>a</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>A</m:mi><m:mo stretchy="false">|</m:mo><m:mo>+</m:mo><m:mo stretchy="false">|</m:mo><m:msub><m:mi>b</m:mi><m:mi>n</m:mi></m:msub><m:mo>−</m:mo><m:mi>B</m:mi><m:mo stretchy="false">|</m:mo><m:mo>&lt;</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac><m:mo>+</m:mo><m:mfrac><m:mi>ε</m:mi><m:mn>2</m:mn></m:mfrac><m:mo>=</m:mo><m:mi>ε</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
 <para id="fs-id1169739110576">□</para>
 <para id="fs-id1169739110579">The algebraic limit laws allow us to evaluate limits for many sequences. For example, consider the sequence <m:math><m:mrow><m:mrow><m:mo>{</m:mo><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:msup><m:mi>n</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:mfrac></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math> As shown earlier, <m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:mn>1</m:mn><m:mtext>/</m:mtext><m:mi>n</m:mi><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>.</m:mo></m:mrow></m:math> Similarly, for any positive integer <m:math><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo></m:mrow></m:math> we can conclude that</para>
@@ -1128,4 +1130,4 @@
 <meaning id="fs-id1169736634914">a sequence that is not bounded is called unbounded</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>
--- "col11963_1.6.1/Calculus/15 Multiple Integration/04 Triple Integrals/index.cnxml.bak"	2019-10-04 14:28:47.207009332 +0100
+++ "col11963_1.6.1/Calculus/15 Multiple Integration/04 Triple Integrals/index.cnxml"	2019-10-04 14:38:20.202600319 +0100
@@ -231,12 +231,12 @@
 <exercise xmlns:data="http://www.w3.org/TR/html5/dom.html#custom-data-attribute" id="fs-id1167794296744"><problem id="fs-id1167794296746">
 <title>Changing the Order of Integration</title>
 <para id="fs-id1167794296751">Consider the iterated integral</para>
-<equation id="fs-id1167794296754" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:mi>y</m:mi><m:mrow/></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
+<equation id="fs-id1167794296754" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup><m:mrow/></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793544547">The order of integration here is first with respect to <emphasis effect="italics">z</emphasis>, then <emphasis effect="italics">y</emphasis>, and then <emphasis effect="italics">x</emphasis>. Express this integral by changing the order of integration to be first with respect to <emphasis effect="italics">x</emphasis>, then <emphasis effect="italics">z</emphasis>, and then <m:math><m:mrow><m:mi>y</m:mi><m:mo>.</m:mo></m:mrow></m:math> Verify that the value of the integral is the same if we let <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>x</m:mi><m:mi>y</m:mi><m:mi>z</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
 </problem>
 <solution id="fs-id1167793612493">
 <para id="fs-id1167793612495">The best way to do this is to sketch the region <m:math><m:mi>E</m:mi></m:math> and its projections onto each of the three coordinate planes. Thus, let</para>
-<equation id="fs-id1167793612502" class="unnumbered"><label/><m:math><m:mrow><m:mi>E</m:mi><m:mo>=</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="false">)</m:mo><m:mo>|</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>x</m:mi><m:mo>≤</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>y</m:mi><m:mo>≤</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>z</m:mi><m:mo>≤</m:mo><m:mi>y</m:mi></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
+<equation id="fs-id1167793612502" class="unnumbered"><label/><m:math><m:mrow><m:mi>E</m:mi><m:mo>=</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi><m:mo stretchy="false">)</m:mo><m:mo>|</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>x</m:mi><m:mo>≤</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>y</m:mi><m:mo>≤</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>,</m:mo><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>z</m:mi><m:mo>≤</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793432448">and</para>
 <equation id="fs-id1167793432451" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mspace width="0.2em"/><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:munderover><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>z</m:mi><m:mo>=</m:mo><m:msup><m:mi>y</m:mi><m:mn>2</m:mn></m:msup></m:mrow></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>z</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>y</m:mi><m:mspace width="0.2em"/><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>=</m:mo><m:mstyle displaystyle="true"><m:mrow><m:munder><m:mo>∭</m:mo><m:mi>E</m:mi></m:munder><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow></m:mstyle><m:mi>d</m:mi><m:mi>V</m:mi><m:mo>.</m:mo></m:mrow></m:math></equation>
 <para id="fs-id1167793424343">We need to express this triple integral as</para>
@@ -795,4 +795,4 @@
 <meaning id="fs-id1167793512095">the triple integral of a continuous function <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>y</m:mi><m:mo>,</m:mo><m:mi>z</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> over a rectangular solid box <m:math><m:mi>B</m:mi></m:math> is the limit of a Riemann sum for a function of three variables, if this limit exists</meaning>
 </definition>
 </glossary>
-</document>
\ No newline at end of file
+</document>