document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=162738>Question 162738</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the finite region enclosed by the curve with equation y=9-x² and the x-axis is rotated through 360º about the x-axis. find, to 3 significant figures, the volume of the solid generated. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #119956 by </B><A HREF=/tutors/aboutme.mpl?userid=vleith><B>vleith(2983)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=vleith><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=119956\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You'll need calculus for this one. If you plot the given function you see that it is a parabola with vertex at (0,9) and x intercepts at (-3,0) and (3,0). So let's find the volume in the solid between x=0 and x=3. Then we just need to double that to get the entire volume\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C+9-x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C+9-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "When you rotate that curve around the x axis, you'll get a solid that you can slice into 'disks'. Imagine the disks are standing on end. As you make the disk thinner and thinner, it approaches becoming a whole bunch of 'circles' with a width of deltax. The radius of each thin circle is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-x%5E2 ALIGN=MIDDLE  ALT=\"9-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from 0 to 3. So the area of each circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%28%289-x%5E2%29%29%5E2 ALIGN=MIDDLE  ALT=\"pi%2A%28%289-x%5E2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%28%289-x%5E2%29%29%5E2 ALIGN=MIDDLE  ALT=\"pi%2A%28%289-x%5E2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%2881-18x%5E2+%2B+x%5E4%29 ALIGN=MIDDLE  ALT=\"pi%2A%2881-18x%5E2+%2B+x%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we need to take that Area and integrate if over from 0 to 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28+A%5Bx%5D%2C+dx%2C+0%2C+3+%29 ALIGN=MIDDLE  ALT=\"int%28+A%5Bx%5D%2C+dx%2C+0%2C+3+%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=int%28+pi%2A%2881-18x%5E2+%2B+x%5E4%29%2C+dx%2C+0%2C+3+%29 ALIGN=MIDDLE  ALT=\"int%28+pi%2A%2881-18x%5E2+%2B+x%5E4%29%2C+dx%2C+0%2C+3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "pi*(81x - 6x^3 + x^5/5) from 0 to 3<BR>\n" );
document.write( "pi*(81*3 - 6*3^3 + (3^5)/5) - pi*(0+0+0)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%28243+-+162+%2B+48.6%29 ALIGN=MIDDLE  ALT=\"pi%2A%28243+-+162+%2B+48.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A129.6 ALIGN=MIDDLE  ALT=\"pi%2A129.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Remember to double it to pick up the volume on the left side of the y-axis<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2A129.6 ALIGN=MIDDLE  ALT=\"2%2Api%2A129.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=814.3 ALIGN=MIDDLE  ALT=\"814.3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=814 ALIGN=MIDDLE  ALT=\"814\"   DISABLED_style= \"max-width:90%; height:auto;\" > to three sig figs </SPAN>\n" );
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