Question 1166024
.
Find the volumes of the solids generated by revolving the regions bounded by the graphs about the given lines: 
y = sqrt(x), y=0, x=3, and x=9 
(a) about the x=axis, 
(b) about the y-axis, 
(c) about the line at x=3, 
(d) and about the line at x=9. 
Use integration and washer/disk method. I have already solved about the x-axis to be 9/2 pi.
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        In the post by @CPhill, part (c) is solved incorrectly.

        I came to bring a correct solution for part (c).



<pre>
        I will use the "shell" method, which suits this problem 
             MUCH BETTER than the "washer/disk" method.



We can present the solid body in this case as a set of thin vertical cylindrical shells 
with the axis of cylindrical shells x=3.


In this case, each shell has the length along y-axis from y=0 to y = {{{sqrt(x)}}} and the radius (x-3), 
so the volume of the solid body is


    V = integral from 3 to 9  of  {{{2pi*(x-3)*sqrt(x)*dx}}},


Integration gives the antiderivative  


       F(x) = 2pi*((2/5)*x^(5/2) - 2*x^(3/2)),  


and we should calculate the difference 


    F(9) - F(3) = [ 2pi*((2/5)*9^(5/2) - 2*9^(3/2)) ] - [ 2pi*((2/5)*3^(5/2)-2*3^(3/2)) ] = 

                = {{{2pi*((2/5)*243 - 2*27)}}} - {{{2pi*((2/5)*9*sqrt(3) - 6*sqrt(3))}}} = 

                = {{{2pi*((216 + 12*sqrt(3))/5)}}} = {{{24pi*((18 + sqrt(3))/5)}}} = {{{24*3.14159265*((18 + sqrt(3))/5)}}} = 297.5523 approximately.


<U>ANSWER</U>.  The volume of the solid is  {{{2pi*((216 + 12*sqrt(3))/5)}}},  or about  297.5523 cubic units, approximately.
</pre>

Solved correctly.


The formula from the @CPhill solution gives the numerical value of


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{(432pi)/5 - (156pi*sqrt(3))/5}}} = {{{(432*3.14159265)/5 - (156*3.14159265*sqrt(3))/5}}} = 101.6620


which is totally wrong.



I'm somewhat surprised by the clumsiness of the instruction. It recommends 
using a washer/disk method, which is ill-suited for this case.
Apparently, the instruction was written by someone with little or no knowledge of the subject.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . 



Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence in solving Math problems is in the experimental stage of development 
and it is far from to be a well-tuned.

It can make mistakes and produce nonsense.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



The @CPhill' solutions are copy-paste &nbsp;Google &nbsp;AI solutions (of its just old outdated version), 
but there is one essential difference.


Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, &nbsp;he &nbsp;NEVER &nbsp;SAYS &nbsp;TRUTH.


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.