document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117974>Question 1117974</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Let R be the region bounded by the curves y =root(x + 1 )and y = (x + 1)^2<BR>\n" );
document.write( ". We will find the<BR>\n" );
document.write( "volume of the solid obtained by rotating R about the horizontal line y = &#8722;1.<BR>\n" );
document.write( "(a) Find the points of intersection of the two curves.<BR>\n" );
document.write( "(b) Sketch a graph of a portion of the curves, the region R, and the line y = &#8722;1.<BR>\n" );
document.write( "(c) Calculate the volume of the solid obtained by rotating R about the horizontal line y = &#8722;1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733262 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733262\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<b>Points of Intersection</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \(x\ +\ 1)^2 =\ \sqrt{x\ +\ 1}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ +\ 1)^4 =\ x\ +\ 1\\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^4\ +\ 4x^3\ +\ 6x^2\ +\ 4x\ +\ 1\ =\ x\ +\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^4\ +\ 4x^3\ +\ 6x^2\ +\ 3x\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x(x^3\ +\ 4x^2\ +\ 6x\ +\ 3)\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x(x\ +\ 1)(x^2\ +\ 3x\ +\ 3)\ =\ 0\">\r<BR>\n" );
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document.write( "Two real roots, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0\ \ \"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ -1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y(0)\ =\ \sqrt{0\ +\ 1}\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y(-1)\ =\ \sqrt{-1\ +\ 1}\ =\ 0\">\r<BR>\n" );
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document.write( "Points of intersection are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,1)\ \ \"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-1,0)\">\r<BR>\n" );
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document.write( "<b>Sketch</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=Volume_of_Rotation.jpg\">\r<BR>\n" );
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document.write( "<b>Volume of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R\"> rotated around <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ -1\"></b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V\ =\ \pi\int_{-1}^0\,\(\sqrt{x\ +\ 1}\ +\ 1\)^2\ -\ \((x\ +\ 1)^2\ +\ 1\)^2\,dx\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ =\ \frac{\pi\(40\(x\ +\ 1\)^{\frac{3}{2}}\ -\ 6x^5\ -\ 30x4\ -\ 80x^3\ -\ 105x^2\ -\ 60x\)}{30}\|_{-1}^0\ =\ \frac{29\pi}{30}\">\r<BR>\n" );
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document.write( "Just slightly north of 3 cubic units.  Verification of the antiderivative is left as an exercise for the student.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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