document.write( "Question 872751: use the given table of values to estimate the volume of the solid formed by revolving
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\n" ); document.write( "y = f(x), $\"0+%3C=+x+%3C=+3\"$ , about the x-axis, \r
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\n" ); document.write( "x | 0 | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |
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\n" ); document.write( "f(x) | 2.3 | 1.4 | 2.9 | 1.1 | 2.4 | 1.6 | 1.9 | \n" ); document.write( "

Algebra.Com's Answer #526432 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Plot the 7 points and connect them:\r\n" );
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document.write( "When we rotate it about the x axis, we'll have a solid figure\r\n" );
document.write( "with this mid-cross section:\r\n" );
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document.write( "It says just to estimate the volume.  So it may be that your teacher\r\n" );
document.write( "just wants you to average up those 7 y-values like this:\r\n" );
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document.write( "and then assume the volume is about the same volume as a cylinder with \r\n" );
document.write( "a radius of  like the cylinder which the green lines \r\n" );
document.write( "represent a mid-cross section of below:\r\n" );
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document.write( "If that's the kind of estimate your teacher wants, then we\r\n" );
document.write( "just use the volume of a cylinder with that average y-value\r\n" );
document.write( "as a its radius.  The cylinder's height h (measured horizontally)\r\n" );
document.write( "is 3 units,\r\n" );
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document.write( "The formula for the volume of a cylinder is\r\n" );
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document.write( "V =  =  = 35.57565167\r\n" );
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document.write( "So that ought to be a pretty good estimate, about 36 cubic units.\r\n" );
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document.write( "Maybe that's all your teacher wants.\r\n" );
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document.write( "Just for fun, let's find the true volume: and compare:\r\n" );
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document.write( "We'll divide that mid-cross section into 6 little trapezoids:\r\n" );
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document.write( "Each of those trapezoids is a mid cross section of a frustrum of\r\n" );
document.write( "a cone, with height h, and the right and left radii are the\r\n" );
document.write( "values of f(x).  The formula for the volume of a frustrum of a\r\n" );
document.write( "cone is:\r\n" );
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document.write( "The height of each one is  so the above will become:\r\n" );
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document.write( "Let the volumes of the frustrums of the cones be V₁,...,V₆\r\n" );
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document.write( "Total volume = \r\n" );
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document.write( "Since all are multiplied by \r\n" );
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document.write( "Total volume = \r\n" );
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document.write( " = \r\n" );
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document.write( "That rounds to 36 cubic units, so the estimate using the cylinder was pretty good.\r\n" );
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document.write( "Edwin

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