document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172398>Question 1172398</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the volume of solid in xyz-space bounded by surfaces y = x^2, y = 2 - x^2, z = 0 and z = y + 3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850822 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=850822\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the volume of the solid.\r<BR>\n" );
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document.write( "**1. Determine the Region in the xy-Plane**\r<BR>\n" );
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document.write( "* We need to find the intersection of the parabolas y = x^2 and y = 2 - x^2.<BR>\n" );
document.write( "* Set them equal: x^2 = 2 - x^2<BR>\n" );
document.write( "* 2x^2 = 2<BR>\n" );
document.write( "* x^2 = 1<BR>\n" );
document.write( "* x = ±1<BR>\n" );
document.write( "* When x = ±1, y = 1.<BR>\n" );
document.write( "* The region in the xy-plane is bounded by these parabolas, with x ranging from -1 to 1.\r<BR>\n" );
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document.write( "**2. Set up the Triple Integral**\r<BR>\n" );
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document.write( "* The volume is given by the triple integral:<BR>\n" );
document.write( "    * V = ∫∫∫ dV<BR>\n" );
document.write( "* The limits of integration are:<BR>\n" );
document.write( "    * z: 0 to y + 3<BR>\n" );
document.write( "    * y: x^2 to 2 - x^2<BR>\n" );
document.write( "    * x: -1 to 1\r<BR>\n" );
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document.write( "* The integral becomes:<BR>\n" );
document.write( "    * V = ∫(from -1 to 1) ∫(from x^2 to 2 - x^2) ∫(from 0 to y + 3) dz dy dx\r<BR>\n" );
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document.write( "**3. Evaluate the Integral**\r<BR>\n" );
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document.write( "* First, integrate with respect to z:<BR>\n" );
document.write( "    * ∫(from 0 to y + 3) dz = [z](from 0 to y + 3) = y + 3<BR>\n" );
document.write( "* Now, integrate with respect to y:<BR>\n" );
document.write( "    * ∫(from x^2 to 2 - x^2) (y + 3) dy = [y^2/2 + 3y](from x^2 to 2 - x^2)<BR>\n" );
document.write( "    * = [(2 - x^2)^2/2 + 3(2 - x^2)] - [(x^2)^2/2 + 3(x^2)]<BR>\n" );
document.write( "    * = [4 - 4x^2 + x^4]/2 + 6 - 3x^2 - x^4/2 - 3x^2<BR>\n" );
document.write( "    * = 2 - 2x^2 + x^4/2 + 6 - 3x^2 - x^4/2 - 3x^2<BR>\n" );
document.write( "    * = 8 - 8x^2\r<BR>\n" );
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document.write( "* Finally, integrate with respect to x:<BR>\n" );
document.write( "    * ∫(from -1 to 1) (8 - 8x^2) dx = [8x - 8x^3/3](from -1 to 1)<BR>\n" );
document.write( "    * = [8(1) - 8(1)^3/3] - [8(-1) - 8(-1)^3/3]<BR>\n" );
document.write( "    * = [8 - 8/3] - [-8 + 8/3]<BR>\n" );
document.write( "    * = 8 - 8/3 + 8 - 8/3<BR>\n" );
document.write( "    * = 16 - 16/3<BR>\n" );
document.write( "    * = (48 - 16)/3<BR>\n" );
document.write( "    * = 32/3\r<BR>\n" );
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document.write( "**Answer**\r<BR>\n" );
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document.write( "The volume of the solid is 32/3 cubic units.<BR>\n" );
document.write( " </SPAN>\n" );
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