document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085933>Question 1085933</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = &#8722;3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700032 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=700032\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/l9zkCxN.png\"><BR>\n" );
document.write( "f(x) = Ln(x) (green curve)<BR>\n" );
document.write( "g(x) = 1 (red horizontal line)<BR>\n" );
document.write( "x = 1 (blue vertical line)<BR>\n" );
document.write( "y = -3 (black dashed line)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Outer radius = R<BR>\n" );
document.write( "R = vertical distance from the red horizontal line to the black dashed horizontal line<BR>\n" );
document.write( "R = 1 - (-3)<BR>\n" );
document.write( "R = 1 + 3<BR>\n" );
document.write( "R = 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Inner radius = r<BR>\n" );
document.write( "r = vertical distance from the green curve to the black dashed horizontal line<BR>\n" );
document.write( "r = Ln(x) - (-3)<BR>\n" );
document.write( "r = Ln(x) + 3<BR>\n" );
document.write( "In contrast to R, the inner radius will change based on x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Find the intersection point between the green curve and the red line<BR>\n" );
document.write( "y = Ln(x)<BR>\n" );
document.write( "1 = Ln(x)<BR>\n" );
document.write( "e^1 = x<BR>\n" );
document.write( "x = e<BR>\n" );
document.write( "The intersection point is (x,y) = (e,1) where e = 2.71828 approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the shaded orange region shown below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/aQvEQ0W.png\"><BR>\n" );
document.write( "represents the region we want to revolve around y = -3 to form the solid of revolution. We're going from a = 1 to b = e\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The integral to set up is therefore\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V = \pi*\int_{a}^{b}\left(\left(R\right)^2 - \left(r\right)^2\right)dx\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V = \pi*\int_{1}^{e}\left(\left(4\right)^2 - \left(Ln(x) + 3\right)^2\right)dx\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V = \pi*\int_{1}^{e}\left(16 - \left(Ln(x) + 3\right)^2\right)dx\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );