document.write( "Question 1208487: The triangle with vertices (a,a), (a, 2a), (2a,2a) is rotated about:
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Algebra.Com's Answer #846931 by ikleyn(52847) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The triangle with vertices (a,a), (a, 2a), (2a,2a) is rotated about:
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document.write( "I will assume that \"a\" is positive real number.\r\n" );
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document.write( "Notice that this given triangle is SPECIAL. It has vertical leg from (a,a), to (a,2a) \r\n" );
document.write( "and horizontal leg from (a,2a) to (2a,2a).  \r\n" );
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document.write( "So, the triangle itself is a right angled triangle. It is \"the upper half of the square\"   a <= x, y <= 2a,\r\n" );
document.write( "cut in two halves by its diagonal y = x.  It facilitates thinking and calculations.\r\n" );
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document.write( "          Let' consider rotation about x-axis.\r\n" );
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document.write( "The volume is the area of the ring in vertical section x= const, integrated over (or along) x-coordinate\r\n" );
document.write( "from  x= a  to  x= 2a.  \r\n" );
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document.write( "The area of the ring is   = .\r\n" );
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document.write( "Thus the volume is the integral of the function    from  \"a\"  to  \"2a\".\r\n" );
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document.write( "The antiderivative is  the function  F(x) = .\r\n" );
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document.write( "So, the final integral value is the difference  F(2a) - F(a), which is\r\n" );
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document.write( "     =  =  = .\r\n" );
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document.write( "Thus the volume of the body of revolution in this case is  ,  in terms of \"a\", assuming \"a\" is positive.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that in case of rotation about y-axis the reasoning is very similar.\r
\n" ); document.write( "\n" ); document.write( "The difference is that the body of rotation is formed in this case by the \" lower triangle of the square \"\r
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\n" ); document.write( "\n" ); document.write( "So, if you got my method above, then it will be easy to you to manage with the other case.\r
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\n" ); document.write( "\n" ); document.write( "Notice that the solution presented by AI under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=The+triangle+with+vertices+(a%2Ca)%2C+(a%2C+2a)%2C+(2a%2C2a)+is+rotated+about+he+x-axis.+Find+the+volume+in+each+case.&rlz=1C1CHBF_enUS1071US1071&oq=The+triangle+with+vertices+(a%2Ca)%2C+(a%2C+2a)%2C+(2a%2C2a)+is+rotated+about+he+x-axis.++Find+the+volume+in+each+case.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRiPAjIHCAIQIRiPAjIHCAMQIRiPAtIBCjE5MjQ1ajBqMTWoAgiwAgE&sourceid=chrome&ie=UTF-8\r
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\n" ); document.write( "\n" ); document.write( "is/was $\"highlight%28highlight%28INCORRECT%29%29\"$ . I just posted my notice about their wrong solution to that address.\r
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\n" ); document.write( "\n" ); document.write( "Now please take my kind advise. The solution is saturated by formulas.\r
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\n" ); document.write( "\n" ); document.write( "THERFORE, do not submit TWO ASSIGNMENTS in ONE POST.\r
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\n" ); document.write( "\n" ); document.write( "Always make it SEPARATELY.\r
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\n" ); document.write( "\n" ); document.write( "ALWAYS submit only one assignment in your post.\r
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\n" ); document.write( "\n" ); document.write( "DO NOT SUBMIT two or more assignments in one post.\r
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\n" ); document.write( "\n" ); document.write( "Doing against this rule, you always be working AGAINST your own interests, \r
\n" ); document.write( "\n" ); document.write( "creating unnecessary difficulties for yourself and for tutors.\r
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