document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006409>Question 1006409</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The two functions are f(x) = x^3 -x and g(x) = 3x I must find the area bounded by both of them. \r<BR>\n" );
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document.write( "First, I set them equal to each other. I get x = 0, 2, -2 \r<BR>\n" );
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document.write( "Therefore, by the addition property of integrals I have [-2,0] and [0,2]. \r<BR>\n" );
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document.write( "Now, I must test points in those intervals to see where the graph is bigger. \r<BR>\n" );
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document.write( "for [-2,0] <BR>\n" );
document.write( "f(-1) = (-1)^3 - (-1) = 0 <BR>\n" );
document.write( "g(-1) = (-1)(3) = - 3 \r<BR>\n" );
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document.write( "so, f is bigger. \r<BR>\n" );
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document.write( "for [0,2] <BR>\n" );
document.write( "f(1) = (1)^3 - 1 = 0 <BR>\n" );
document.write( "g(1) = 3(1) = 3 \r<BR>\n" );
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document.write( "so, g is bigger \r<BR>\n" );
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document.write( "Now, to set up the integrals \r<BR>\n" );
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document.write( "int[from -2 to 0] (f(x)-g(x))dx + int[from 0 to 2] (g(x)-f(x))dx \r<BR>\n" );
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document.write( "integrating the first I get: <BR>\n" );
document.write( "(x^4)/4 -(4x^2)/2 \r<BR>\n" );
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document.write( "integrating the second I get: <BR>\n" );
document.write( "(3x^2)/2 - (x^4)/4 + (x^2)/2 \r<BR>\n" );
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document.write( "evaluating the first I get the value -4 <BR>\n" );
document.write( "evaluating the second I get the value 0 \r<BR>\n" );
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document.write( "Area = |-4| = 4? \r<BR>\n" );
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document.write( "Something seems off. \r<BR>\n" );
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document.write( "Please help <BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622558 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=622558\">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=f%28x%29-g%28x%29=x%5E3-x-3x=x%5E3-4x ALIGN=MIDDLE  ALT=\"f%28x%29-g%28x%29=x%5E3-x-3x=x%5E3-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29-f%28x%29=-%28f%28x%29-g%28x%29%29=-x%5E3%2B4x ALIGN=MIDDLE  ALT=\"g%28x%29-f%28x%29=-%28f%28x%29-g%28x%29%29=-x%5E3%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The integrals are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28f%28x%29-g%28x%29%29%2C+dx%2C+-2%2C+0+%29=int%28%28x%5E3-4x%29%2C+dx%2C+-2%2C+0+%29=%28matrix%283%2C1%2C%22%22%2Cx%5E4%2F4-2x%5E2%2C%22%22%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2830%2C90%2C-1%2C-0.6%2C-1%2C1%2C%0D%0Aline%28-0.95%2C-0.95%2C-0.95%2C0.95%29%2Clocate%28-0.65%2C0.1%2C%22=%22%29%2C%0D%0Alocate%28-0.9%2C0.93%2C0%29%2Clocate%28-0.9%2C-0.73%2C-2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%5E4%2F4-2%2A0%5E2%29-%28%28-2%29%5E4%2F4-2%28-2%29%5E2%29=0-%2816%2F4-2%2A4%29=-%284-8%29=-%28-4%29=4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28f%28x%29-g%28x%29%29%2C+dx%2C+0%2C+2+%29=int%28%28-x%5E3%2B4x%29%2C+dx%2C+-2%2C+0+%29=%28matrix%283%2C1%2C%22%22%2C-x%5E4%2F4%2B2x%5E2%2C%22%22%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%2830%2C90%2C-1%2C-0.6%2C-1%2C1%2C%0D%0Aline%28-0.95%2C-0.95%2C-0.95%2C0.95%29%2Clocate%28-0.65%2C0.1%2C%22=%22%29%2C%0D%0Alocate%28-0.9%2C0.93%2C0%29%2Clocate%28-0.9%2C-0.73%2C-2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%5E4%2F4%2B2%2A2%5E2%29-%28-0%5E4%2F4%2B2%2A0%5E2%29=%28-16%2F4%2B2%2A4%29-0=-4%2B8=4 ALIGN=MIDDLE  ALT=\"%28-2%5E4%2F4%2B2%2A2%5E2%29-%28-0%5E4%2F4%2B2%2A0%5E2%29=%28-16%2F4%2B2%2A4%29-0=-4%2B8=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both integrals are the same, but that should be no surprise,<BR>\n" );
document.write( "because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd function,<BR>\n" );
document.write( "meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-f%28x%29+for+all+values+of+%7B%7B%7Bx ALIGN=MIDDLE  ALT=\"f%28-x%29=-f%28x%29+for+all+values+of+%7B%7B%7Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that makes its graph symmetrical with respect to the origin,<BR>\n" );
document.write( "meaning that its graph rotated <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > falls on to of the original graph.<BR>\n" );
document.write( "The same can be said of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and of course, the same can be said of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-g%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29-g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-3%2C3%2C-8%2C8%2Cx%5E3-x%2C3x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-3%2C3%2C-8%2C8%2Cx%5E3-x%2C3x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );