document.write( "Question 1166024: Find the volumes of the solids generated by revolving the regions bounded by the graphs about the given lines: y=sqrtx, y=0, x=3, and x=9 about the x=axis, the y-axis, the line at x=3, and the line at x=9. Use integration and washer/disk method. I have already solved about the x-axis to be 9/2 pi. \n" ); document.write( "

Algebra.Com's Answer #852905 by n2(19) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Find the volumes of the solids generated by revolving the regions bounded by the graphs about the given lines:
\n" ); document.write( "y = sqrt(x), y=0, x=3, and x=9
\n" ); document.write( "(a) about the x=axis,
\n" ); document.write( "(b) about the y-axis,
\n" ); document.write( "(c) about the line at x=3,
\n" ); document.write( "(d) and about the line at x=9.
\n" ); document.write( "Use integration and washer/disk method. I have already solved about the x-axis to be 9/2 pi.
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\n" ); document.write( "\n" ); document.write( " In the post by @CPhill, part (c) is solved incorrectly.\r
\n" ); document.write( "\n" ); document.write( " I came to bring a correct solution for part (c).\r
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document.write( "        I will use the \"shell\" method, which suits this problem \r\n" );
document.write( "             MUCH BETTER than the \"washer/disk\" method.\r\n" );
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document.write( "We can present the solid body in this case as a set of thin vertical cylindrical shells \r\n" );
document.write( "with the axis of cylindrical shells x=3.\r\n" );
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document.write( "In this case, each shell has the length along y-axis from y=0 to y =  and the radius (x-3), \r\n" );
document.write( "so the volume of the solid body is\r\n" );
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document.write( "    V = integral from 3 to 9  of  ,\r\n" );
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document.write( "Integration gives the antiderivative  \r\n" );
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document.write( "       F(x) = 2pi*((2/5)*x^(5/2) - 2*x^(3/2)),  \r\n" );
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document.write( "and we should calculate the difference \r\n" );
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document.write( "    F(9) - F(3) = [ 2pi*((2/5)*9^(5/2) - 2*9^(3/2)) ] - [ 2pi*((2/5)*3^(5/2)-2*3^(3/2)) ] = \r\n" );
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document.write( "                =  -  = \r\n" );
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document.write( "                =  =  =  = 297.5523 approximately.\r\n" );
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document.write( "ANSWER.  The volume of the solid is  ,  or about  297.5523 cubic units, approximately.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved correctly.\r
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\n" ); document.write( "\n" ); document.write( "The formula from the @CPhill solution gives the numerical value of\r
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\n" ); document.write( "\n" ); document.write( "          $\"%28432pi%29%2F5+-+%28156pi%2Asqrt%283%29%29%2F5\"$ = $\"%28432%2A3.14159265%29%2F5+-+%28156%2A3.14159265%2Asqrt%283%29%29%2F5\"$ = 101.6620\r
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\n" ); document.write( "\n" ); document.write( "which is totally wrong.\r
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\n" ); document.write( "\n" ); document.write( "I'm somewhat surprised by the clumsiness of the instruction. It recommends
\n" ); document.write( "using a washer/disk method, which is ill-suited for this case.
\n" ); document.write( "Apparently, the instruction was written by someone with little or no knowledge of the subject.\r
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence in solving Math problems is in the experimental stage of development
\n" ); document.write( "and it is far from to be a well-tuned.\r
\n" ); document.write( "\n" ); document.write( "It can make mistakes and produce nonsense.\r
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again, it made an error.\r
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\n" ); document.write( "\n" ); document.write( "The @CPhill' solutions are copy-paste Google AI solutions (of its just old outdated version),
\n" ); document.write( "but there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
\n" ); document.write( "and can make errors/mistakes.\r
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.
\n" ); document.write( "So, he NEVER SAYS TRUTH.\r
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\n" ); document.write( "\n" ); document.write( "Every time, @CPhill embarrassed to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth, as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it, their reliability is ZERO and their creadability is ZERO, too.\r
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