document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1101661>Question 1101661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the area of the region bounded by the graph of y=x^3, the line x=2 and the x-axis is revolved about the line x=-1 then find the volume of the solid revolution.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #716494 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=716494\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I was hoping another tutor would answer this question, so that I could try to solve it and compare answers.  I often have trouble setting up this kind of problem correctly, so I was glad to see another solution.<br><BR>\n" );
document.write( "I did come up with the same answer as the other tutor, but by a different method.<br><BR>\n" );
document.write( "So you can look at the two methods and see if you have a preference for one or the other.<br><BR>\n" );
document.write( "The other tutor used the shell method in his solution.  I generally have better luck with the disk method; so that is what I used.<br><BR>\n" );
document.write( "And so that I could visualize the problem better, I found the answer by finding the volume as the volume of a cylinder, minus the volume of the solid of revolution between the curve and the line y=8, instead of finding the volume of the solid of revolution between the curve and the line y=0 (i.e., the x-axis).<br><BR>\n" );
document.write( "So I'm integrating in the y direction from 0 to 8; and my disk has volume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ar%5E2%2Ady ALIGN=MIDDLE  ALT=\"pi%2Ar%5E2%2Ady\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The curve is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E3 ALIGN=MIDDLE  ALT=\"y+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >; since I'm integrating in the y direction, I convert this to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+y%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"x+=+y%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the rotation is about the line x = -1, the radius r of my disk is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F3%29%2B1 ALIGN=MIDDLE  ALT=\"y%5E%281%2F3%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the volume of my disk is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%28y%5E%281%2F3%29%2B1%29%5E2%2Ady ALIGN=MIDDLE  ALT=\"pi%2A%28y%5E%281%2F3%29%2B1%29%5E2%2Ady\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%28y%5E%282%2F3%29%2B2y%5E%281%2F3%29%2B1%29%2Ady ALIGN=MIDDLE  ALT=\"pi%2A%28y%5E%282%2F3%29%2B2y%5E%281%2F3%29%2B1%29%2Ady\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Doing the integration, the volume of my solid of revolution is then pi times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F5%29y%5E%285%2F3%29%2B%283%2F2%29y%5E%284%2F3%29%2By ALIGN=MIDDLE  ALT=\"%283%2F5%29y%5E%285%2F3%29%2B%283%2F2%29y%5E%284%2F3%29%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" > evaluated between 0 and 8.<br><BR>\n" );
document.write( "For y=8: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F5%29%2A32%2B%283%2F2%29%2A16+%2B+8+=+256%2F5 ALIGN=MIDDLE  ALT=\"%283%2F5%29%2A32%2B%283%2F2%29%2A16+%2B+8+=+256%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and of course 0 for y=0.<br><BR>\n" );
document.write( "The volume of my cylinder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%283%5E2%29%288%29+=+72%2Api ALIGN=MIDDLE  ALT=\"pi%283%5E2%29%288%29+=+72%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the volume we are looking for in the problem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%2Api+-+%28256%2F5%29%2Api+=+%28104%2F5%29%2Api ALIGN=MIDDLE  ALT=\"72%2Api+-+%28256%2F5%29%2Api+=+%28104%2F5%29%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "... which is the answer the other tutor got. </SPAN>\n" );
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