document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=345462>Question 345462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation: Let A be the area of the region that lies under the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+2x-x%5E2+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+2x-x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, above the x-asis and between x=0 and x=1.<BR>\n" );
document.write( "a. Estimate A by taking the sample points to be right endpoints and using three subintervals.<BR>\n" );
document.write( "b. Using right endpoints, find an expression for the area as a limit. Evaluate the limit. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #247143 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=247143\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Break up the interval into four segments.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA%2Ax=%281-0%29%2F4=1%2F4 ALIGN=MIDDLE  ALT=\"DELTA%2Ax=%281-0%29%2F4=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%281%2F4%29-%281%2F4%29%5E2=1%2F2-1%2F16=8%2F16-1%2F16=7%2F16 ALIGN=MIDDLE  ALT=\"y=2%281%2F4%29-%281%2F4%29%5E2=1%2F2-1%2F16=8%2F16-1%2F16=7%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F2 ALIGN=MIDDLE  ALT=\"x=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%281%2F2%29-%281%2F2%29%5E2=1-1%2F4=3%2F4 ALIGN=MIDDLE  ALT=\"y=2%281%2F2%29-%281%2F2%29%5E2=1-1%2F4=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F4 ALIGN=MIDDLE  ALT=\"x=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%283%2F4%29-%283%2F4%29%5E2=24%2F16-9%2F16=15%2F16 ALIGN=MIDDLE  ALT=\"y=2%283%2F4%29-%283%2F4%29%5E2=24%2F16-9%2F16=15%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%281%29-1%5E2=2 ALIGN=MIDDLE  ALT=\"y=2%281%29-1%5E2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=DELTA%2Ax%2A%28y%281%2F4%29%2By%281%2F2%29%2By%283%2F4%29%2By%281%29%29 ALIGN=MIDDLE  ALT=\"A=DELTA%2Ax%2A%28y%281%2F4%29%2By%281%2F2%29%2By%283%2F4%29%2By%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F4%29%2A%287%2F16%2B3%2F4%2B15%2F16%2B2%29 ALIGN=MIDDLE  ALT=\"A=%281%2F4%29%2A%287%2F16%2B3%2F4%2B15%2F16%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F4%29%2A%287%2F16%2B12%2F16%2B15%2F16%2B32%2F16%29 ALIGN=MIDDLE  ALT=\"A=%281%2F4%29%2A%287%2F16%2B12%2F16%2B15%2F16%2B32%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F4%29%2A%287%2F16%2B12%2F16%2B15%2F16%2B32%2F16%29 ALIGN=MIDDLE  ALT=\"A=%281%2F4%29%2A%287%2F16%2B12%2F16%2B15%2F16%2B32%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2F4%29%2A%2866%2F16%29 ALIGN=MIDDLE  ALT=\"A=%281%2F4%29%2A%2866%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A=33%2F32%29 ALIGN=MIDDLE  ALT=\"highlight%28A=33%2F32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
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document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx=1%2Fn ALIGN=MIDDLE  ALT=\"dx=1%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=dx%2Asum%28y%5Bi%5D%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A=dx%2Asum%28y%5Bi%5D%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2Fn%29%2Asum%28%282%2Ax%5Bi%5D-x%5Bi%5D%5E2%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A=%281%2Fn%29%2Asum%28%282%2Ax%5Bi%5D-x%5Bi%5D%5E2%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%281%2Fn%29%2Asum%28%282%2Ax%5Bi%5D%29%2Ci=1%2Cn%29-%281%2Fn%29%2Asum%28%28x%5Bi%5D%5E2%29%2Ci=1%2Cn%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=A%5B1%5D%2BA%5B2%5D ALIGN=MIDDLE  ALT=\"A=A%5B1%5D%2BA%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let's work on each sum separately,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%281%2Fn%29%2Asum%28%282%2Ax%5Bi%5D%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%281%2Fn%29%2Asum%28%282%2Ax%5Bi%5D%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D=0%2Bi%2Fn ALIGN=MIDDLE  ALT=\"x%5Bi%5D=0%2Bi%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D=i%2Fn ALIGN=MIDDLE  ALT=\"x%5Bi%5D=i%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D%5E2=i%5E2%2Fn%5E2 ALIGN=MIDDLE  ALT=\"x%5Bi%5D%5E2=i%5E2%2Fn%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%281%2Fn%29%2Asum%28%282%2A%28i%2Fn%29%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%281%2Fn%29%2Asum%28%282%2A%28i%2Fn%29%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%282%2Fn%5E2%29%2Asum%28%28i%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%282%2Fn%5E2%29%2Asum%28%28i%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%282%2Fn%5E2%29%2A%28n%28n%2B1%29%2F2%29 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%282%2Fn%5E2%29%2A%28n%28n%2B1%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%28n%28n%2B1%29%29%2Fn%5E2 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%28n%28n%2B1%29%29%2Fn%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B1%5D=%28n%5E2%2B1%29%2Fn%5E2 ALIGN=MIDDLE  ALT=\"A%5B1%5D=%28n%5E2%2B1%29%2Fn%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A%5B1%5D=1%2B1%2Fn%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28A%5B1%5D=1%2B1%2Fn%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%281%2Fn%29%2Asum%28%28x%5Bi%5D%5E2%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%281%2Fn%29%2Asum%28%28x%5Bi%5D%5E2%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%281%2Fn%29%2Asum%28%28%28i%2Fn%29%5E2%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%281%2Fn%29%2Asum%28%28%28i%2Fn%29%5E2%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%281%2Fn%5E3%29%2Asum%28%28i%5E2%29%2Ci=1%2Cn%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%281%2Fn%5E3%29%2Asum%28%28i%5E2%29%2Ci=1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%281%2Fn%5E3%29%2A%28%28n%28n%2B1%29%282n%2B1%29%29%2F6%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%281%2Fn%5E3%29%2A%28%28n%28n%2B1%29%282n%2B1%29%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%281%2Fn%5E2%29%2A%28%282n%5E2%2B3n%2B1%29%2F6%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%281%2Fn%5E2%29%2A%28%282n%5E2%2B3n%2B1%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=%282n%5E2%2B3n%2B1%29%2F%286n%5E2%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=%282n%5E2%2B3n%2B1%29%2F%286n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5B2%5D=2%2F6%2B3%2F%286n%29%2B1%2F%286n%5E2%29 ALIGN=MIDDLE  ALT=\"A%5B2%5D=2%2F6%2B3%2F%286n%29%2B1%2F%286n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A%5B2%5D=1%2F3%2B1%2F%282n%29%2B1%2F%286n%5E2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28A%5B2%5D=1%2F3%2B1%2F%282n%29%2B1%2F%286n%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now put it back together,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=A%5B1%5D-A%5B2%5D ALIGN=MIDDLE  ALT=\"A=A%5B1%5D-A%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1%2B1%2Fn%5E2-%281%2F3%2B1%2F%282n%29%2B1%2F%286n%5E2%29%29 ALIGN=MIDDLE  ALT=\"A=1%2B1%2Fn%5E2-%281%2F3%2B1%2F%282n%29%2B1%2F%286n%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1-1%2F3-1%2F%282n%29%2B1%2Fn%5E2-1%2F%286n%5E2%29 ALIGN=MIDDLE  ALT=\"A=1-1%2F3-1%2F%282n%29%2B1%2Fn%5E2-1%2F%286n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2F3-1%2F%282n%29%2B5%2F%286n%5E2%29 ALIGN=MIDDLE  ALT=\"A=2%2F3-1%2F%282n%29%2B5%2F%286n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the limit, as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2%2F3-0%2B0 ALIGN=MIDDLE  ALT=\"A=2%2F3-0%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28A=2%2F3%29 ALIGN=MIDDLE  ALT=\"highlight_green%28A=2%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );