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math05.htm
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<td width=614 valign=top style='width:460.7pt;border:solid blue 2.25pt;
border-left:none;padding:0cm 5.4pt 0cm 5.4pt'>
<p class=MsoNormal align=center style='margin-bottom:0cm;margin-bottom:.0001pt;
text-align:center;line-height:150%'><b><span style='font-size:14.0pt;
line-height:150%;font-family:"Bookman Old Style",serif;color:blue'><a
href="math.htm">MATHEMATICS
EXTENSION 2</a></span></b></p>
<p class=MsoNormal align=center style='margin-bottom:0cm;margin-bottom:.0001pt;
text-align:center;line-height:150%'><b><span style='font-size:14.0pt;
line-height:150%;font-family:"Bookman Old Style",serif;color:blue'>4 UNIT
MATHEMATICS</span></b></p>
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text-align:center;line-height:150%'><b><span style='font-size:14.0pt;
line-height:150%;font-family:"Bookman Old Style",serif;color:#E46C0A'>TOPIC
5: VOLUMES</span></b></p>
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border-left:none;padding:0cm 5.4pt 0cm 5.4pt'>
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<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><span
style='font-size:12.0pt;line-height:115%'>This is only a short topic on a
practical example on the use of the definite integral to calculate the volume
of a three-dimensional object. You need to have good skills in sketching
curves and integrating functions to be successful in answering examination
questions on volume calculations. The best approach in answering volume
questions is to work from first principles and not start from an equation. <o:p></o:p></span></p>
</td>
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text-align:right;line-height:normal'><b><span style='font-size:12.0pt;
font-family:"Bookman Old Style",serif;color:blue'>TOPICS</span></b><span
style='font-size:12.0pt'><o:p></o:p></span></p>
</td>
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mso-border-left-themecolor:background1;padding:0cm 5.4pt 0cm 5.4pt'>
<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><b><span
style='font-size:12.0pt;line-height:115%'><a
href="vol5_01.pdf">5.1<span
style='mso-spacerun:yes'> </span>Area under a curve</a></span></b><span
style='font-size:12.0pt;line-height:115%'><o:p></o:p></span></p>
<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><span
style='font-size:12.0pt;line-height:115%'>The area under a curve </span><i><span
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style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman",serif'>
= <i>f</i>(<i>x</i>)</span><span style='font-size:12.0pt;line-height:115%'> between
the limits </span><i><span style='font-size:12.0pt;line-height:115%;
font-family:"Times New Roman",serif'>x</span></i><sub><span style='font-size:
12.0pt;line-height:115%;font-family:"Times New Roman",serif'>1</span></sub><span
style='font-size:12.0pt;line-height:115%'> and </span><i><span
style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman",serif'>x</span></i><sub><span
style='font-size:12.0pt;line-height:115%;font-family:"Times New Roman",serif'>2</span></sub><span
style='font-size:12.0pt;line-height:115%'> can be found by the evaluation of
a definite integral.<o:p></o:p></span></p>
<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><span
style='font-size:12.0pt;line-height:115%'> <o:p></o:p></span></p>
<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><b
style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;line-height:
115%'><a href="vol5_02.pdf">5.2<span
style='mso-spacerun:yes'> </span>Volumes based upon cross sections</a><o:p></o:p></span></b></p>
<p class=MsoNormal style='margin-bottom:0cm;margin-bottom:.0001pt'><span
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