document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934217>Question 934217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the area of the region bounded by functions:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29=+-x%5E2+%2B+3x+ ALIGN=MIDDLE  ALT=\"+f%28x%29=+-x%5E2+%2B+3x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28x%29+=+2x%5E3+-+x%5E2+-+5x+ ALIGN=MIDDLE  ALT=\"+g%28x%29+=+2x%5E3+-+x%5E2+-+5x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #567319 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=567319\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28%22f%28x%29%22=+-x%5E2+%2B+3x%2C+%22g%28x%29%22=2x%5E3+-+x%5E2+-+5x%29 ALIGN=MIDDLE  ALT=\"system%28%22f%28x%29%22=+-x%5E2+%2B+3x%2C+%22g%28x%29%22=2x%5E3+-+x%5E2+-+5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We find where the two curves intersect by setting them equal\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2+%2B+3x=2x%5E3+-+x%5E2+-+5x ALIGN=MIDDLE  ALT=\"-x%5E2+%2B+3x=2x%5E3+-+x%5E2+-+5x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=2x%5E3-8x ALIGN=MIDDLE  ALT=\"0=2x%5E3-8x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-8x=0 ALIGN=MIDDLE  ALT=\"2x%5E3-8x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-4x=0 ALIGN=MIDDLE  ALT=\"x%5E3-4x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2-4%29=0 ALIGN=MIDDLE  ALT=\"x%28x%5E2-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-2%29%28x%2B2%29=0 ALIGN=MIDDLE  ALT=\"x%28x-2%29%28x%2B2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Three solutions (points of intersection):\r<BR>\n" );
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document.write( "x=0; x-2=0; x+2=0<BR>\n" );
document.write( "       x=2;   x=-2\r<BR>\n" );
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document.write( "The curves intersect at (-2,-10), (0,0), (2,2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28640%2F3%2C400%2C-4%2C4%2C-12%2C3%2Cgraph%28640%2F3%2C400%2C-4%2C4%2C-12%2C3%2C2x%5E3+-+x%5E2+-+5x%29%2C+%0D%0Agraph%28640%2F3%2C400%2C-4%2C4%2C-12%2C3%2C-x%5E2+%2B+3x%29%2C%0D%0Alocate%28-4%2C-10%2C%22%28-2%2C-10%29%22%29%2C+locate%280%2C0%2C%22%280%2C0%29%22%29%2Clocate%282.3%2C2.4%2C%22%282%2C2%29%22%29+%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "As we see, there are two loops bound by these curves.<BR>\n" );
document.write( "We must find the areas of the two loops separately, because<BR>\n" );
document.write( "the upper curve in the left loop is the lower curve in the<BR>\n" );
document.write( "loop on the right.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area ALIGN=MIDDLE  ALT=\"Area\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%0D%0A%0D%0A%28%28matrix%284%2C1%2Cexpression%2Cfor%2Cupper%2Ccurve%29%29%5E%22%22-%28matrix%284%2C1%2Cexpression%2Cfor%2Clower%2Ccurve%29%29%29%2Cdx%2C%0D%0A%0D%0Amatrix%281%2C2%2Cleast%2C%22x-value%22%29%2Cmatrix%281%2C2%2Chighest%2C%22x-value%22%29%0D%0A%0D%0A%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Left loop:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28%282x%5E3-x%5E2-5x%29-%28-x%5E2+%2B+3x%29%5E%22%22%29%2Cdx%2C-2%2C0%29 ALIGN=MIDDLE  ALT=\"int%28%28%282x%5E3-x%5E2-5x%29-%28-x%5E2+%2B+3x%29%5E%22%22%29%2Cdx%2C-2%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=int%28%282x%5E3-x%5E2-5x%2Bx%5E2+-+3x%29%2Cdx%2C-2%2C0%29 ALIGN=MIDDLE  ALT=\"int%28%282x%5E3-x%5E2-5x%2Bx%5E2+-+3x%29%2Cdx%2C-2%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=int%28%282x%5E3-8x%29%2Cdx%2C-2%2C0%29 ALIGN=MIDDLE  ALT=\"int%28%282x%5E3-8x%29%2Cdx%2C-2%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=%0D%0A%0D%0Amatrix%283%2C3%2C%0D%0A++%22%22%2C++++++%22%7C%22%2C0%2C%0D%0Ax%5E4%2F2-4x%5E2%2C%22%7C%22%2C%22%22%2C%0D%0A++%22%22%2C++++++%22%7C%22%2C-2%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%5E4%2F2-4%2A0%5E2%29-%28%28-2%29%5E4%2F2-4%2A%28-2%29%5E2%29+ ALIGN=MIDDLE  ALT=\"%280%5E4%2F2-4%2A0%5E2%29-%28%28-2%29%5E4%2F2-4%2A%28-2%29%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Right loop:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28%28-x%5E2+%2B+3x%29-%282x%5E3-x%5E2-5x%29%5E%22%22%29%2Cdx%2C0%2C2%29 ALIGN=MIDDLE  ALT=\"int%28%28%28-x%5E2+%2B+3x%29-%282x%5E3-x%5E2-5x%29%5E%22%22%29%2Cdx%2C0%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=int%28%28-x%5E2%2B3x-2x%5E3%2Bx%5E2%2B5x%29%2Cdx%2C0%2C2%29 ALIGN=MIDDLE  ALT=\"int%28%28-x%5E2%2B3x-2x%5E3%2Bx%5E2%2B5x%29%2Cdx%2C0%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=int%28%28-2x%5E3%2B8x%29%2Cdx%2C0%2C2%29 ALIGN=MIDDLE  ALT=\"int%28%28-2x%5E3%2B8x%29%2Cdx%2C0%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   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=%0D%0A%0D%0Amatrix%283%2C3%2C%0D%0A++%22%22%2C++++++%22%7C%22%2C2%2C%0D%0A-x%5E4%2F2%2B4x%5E2%2C%22%7C%22%2C%22%22%2C%0D%0A++%22%22%2C++++++%22%7C%22%2C0%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-%282%29%5E4%2F2%2B4%2A%282%29%5E2%29%28-0%5E4%2F2%2B4%2A0%5E2%29+ ALIGN=MIDDLE  ALT=\"%28-%282%29%5E4%2F2%2B4%2A%282%29%5E2%29%28-0%5E4%2F2%2B4%2A0%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "Total area = 8+8 = 16\r<BR>\n" );
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document.write( "Edwin</pre> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );