document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177659>Question 1177659</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Background info: In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered in this unit. Sketch the graph of the function y = 20x − x2, and approximate the area under the curve in the interval [0, 20] by dividing the area into the given numbers of rectangles.\r<BR>\n" );
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document.write( "Question: Use 10 rectangles to approximate the area under the curve.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #806770 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=806770\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sketch the graph of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+20x-x%5E2 ALIGN=MIDDLE  ALT=\"y+=+20x-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+600%2C+600%2C+-10%2C+35%2C+-10%2C+110%2C+20x-x%5E2%29+ ALIGN=MIDDLE  ALT=\"graph%28+600%2C+600%2C+-10%2C+35%2C+-10%2C+110%2C+20x-x%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "divide the area in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > rectangles \r<BR>\n" );
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document.write( "In order to figure the width of each rectangle we can use the following formula:\r<BR>\n" );
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document.write( " &Delta; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28b-a%29%2Fn ALIGN=MIDDLE  ALT=\"x=%28b-a%29%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0 ALIGN=MIDDLE  ALT=\"a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b=20 ALIGN=MIDDLE  ALT=\"+b=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10 ALIGN=MIDDLE  ALT=\"n=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we get:\r<BR>\n" );
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document.write( "&Delta; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2820-0%29%2F10=2 ALIGN=MIDDLE  ALT=\"x=%2820-0%29%2F10=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=each ALIGN=MIDDLE  ALT=\"each\"   DISABLED_style= \"max-width:90%; height:auto;\" > rectangle must have a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width ALIGN=MIDDLE  ALT=\"width\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+ ALIGN=MIDDLE  ALT=\"2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >units\r<BR>\n" );
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document.write( "We can now calculate the height of each rectangle. So we figure the y-value of each corner of the rectangles. We get the following heights:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B1%5D=36 ALIGN=MIDDLE  ALT=\"h%5B1%5D=36\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2+ ALIGN=MIDDLE  ALT=\"x=2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >units-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+20%2A2-2%5E2=36 ALIGN=MIDDLE  ALT=\"y+=+20%2A2-2%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B2%5D=64 ALIGN=MIDDLE  ALT=\"h%5B2%5D=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4+ ALIGN=MIDDLE  ALT=\"x=4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >units-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+20%2A4-4%5E2=64 ALIGN=MIDDLE  ALT=\"y+=+20%2A4-4%5E2=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B3%5D=84 ALIGN=MIDDLE  ALT=\"h%5B3%5D=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >.............<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6+ ALIGN=MIDDLE  ALT=\"x=6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >units-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+20%2A6-6%5E2=84 ALIGN=MIDDLE  ALT=\"y+=+20%2A6-6%5E2=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >...and so on\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B4%5D=+96 ALIGN=MIDDLE  ALT=\"h%5B4%5D=+96\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B5%5D=100 ALIGN=MIDDLE  ALT=\"h%5B5%5D=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B6%5D=96 ALIGN=MIDDLE  ALT=\"h%5B6%5D=96\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B7%5D=84 ALIGN=MIDDLE  ALT=\"h%5B7%5D=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B8%5D=64 ALIGN=MIDDLE  ALT=\"h%5B8%5D=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B9%5D=36 ALIGN=MIDDLE  ALT=\"h%5B9%5D=36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B10%5D=0 ALIGN=MIDDLE  ALT=\"h%5B10%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so now we can use the following formula to find the area under the graph. Basically what the formula does is add the areas of the rectangles:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=+2h%5B1%5D%2B2h%5B2%5D%2B2h%5B3%5D%2B2h%5B4%5D%2B2h%5B5%5D%2B2h%5B6%5D%2B2h%5B7%5D%2B2h%5B8%5D%2B2h%5B9%5D%2B2h%5B10%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2836%2B64%2B84%2B96%2B100%2B96%2B84%2B64%2B36%2B0%29 ALIGN=MIDDLE  ALT=\"A=2%2836%2B64%2B84%2B96%2B100%2B96%2B84%2B64%2B36%2B0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1320 ALIGN=MIDDLE  ALT=\"A=1320\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> approximate the area under the curve in the interval [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >] \r<BR>\n" );
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