document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182678>Question 1182678</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The area bounded by y = 3, x = 2, y = -3 and x = 0 is revolved about the y-axis.<BR>\n" );
document.write( "The moment of inertia of the solid is </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849911 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849911\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the moment of inertia of the solid:\r<BR>\n" );
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document.write( "1. **Describe the shape:** The region bounded by y = 3, x = 2, y = -3, and x = 0 is a rectangle. When revolved about the y-axis, this rectangle forms a hollow cylinder (or a cylindrical shell).\r<BR>\n" );
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document.write( "2. **Dimensions of the cylinder:**<BR>\n" );
document.write( "    * Inner radius (r₁): 0<BR>\n" );
document.write( "    * Outer radius (r₂): 2<BR>\n" );
document.write( "    * Height (h): 3 - (-3  We are not given a density, so we'll assume a uniform density (ρ). Then, we will find the moment of inertia *in terms of ρ*. If you're given a density, you would multiply by it at the end to get a numerical moment of inertia.\r<BR>\n" );
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document.write( "4. **Moment of Inertia of a hollow cylinder:** The moment of inertia (I) of a hollow cylinder about its central axis is given by the formula:\r<BR>\n" );
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document.write( "   I = (1/2) * m * (r₁² + r₂²)\r<BR>\n" );
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document.write( "   where m is the mass of the cylinder.\r<BR>\n" );
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document.write( "5. **Mass in terms of density and volume:** The mass of the cylinder is equal to the product of its density and volume.  <BR>\n" );
document.write( "    * The volume of the hollow cylinder is given by V = πh(r₂² - r₁²)<BR>\n" );
document.write( "    * Volume: V = π * 6 * (2² - 0²) = 24π<BR>\n" );
document.write( "    * Thus, the mass m = ρ * V = 24πρ\r<BR>\n" );
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document.write( "6. **Substitute and calculate:**\r<BR>\n" );
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document.write( "   I = (1/2) * (24πρ) * (0² + 2²)<BR>\n" );
document.write( "   I = 12πρ * 4<BR>\n" );
document.write( "   I = 48πρ\r<BR>\n" );
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document.write( "Therefore, the moment of inertia of the solid is $\boxed{48\pi\rho}$.  If the density (ρ) were given, you would multiply by that value to find the numerical moment of inertia.<BR>\n" );
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