<PRE><B>Question 1101661</B>
&lt;br&gt;
I was hoping another tutor would answer this question, so that I could try to solve it and compare answers.  I often have trouble setting up this kind of problem correctly, so I was glad to see another solution.&lt;br&gt;
I did come up with the same answer as the other tutor, but by a different method.&lt;br&gt;
So you can look at the two methods and see if you have a preference for one or the other.&lt;br&gt;
The other tutor used the shell method in his solution.  I generally have better luck with the disk method; so that is what I used.&lt;br&gt;
And so that I could visualize the problem better, I found the answer by finding the volume as the volume of a cylinder, minus the volume of the solid of revolution between the curve and the line y=8, instead of finding the volume of the solid of revolution between the curve and the line y=0 (i.e., the x-axis).&lt;br&gt;
So I&#39;m integrating in the y direction from 0 to 8; and my disk has volume {{{pi*r^2*dy}}}.&lt;br&gt;
The curve is {{{y = x^3}}}; since I&#39;m integrating in the y direction, I convert this to {{{x = y^(1/3)}}}&lt;br&gt;
Since the rotation is about the line x = -1, the radius r of my disk is {{{y^(1/3)+1}}}&lt;br&gt;
So the volume of my disk is {{{pi*(y^(1/3)+1)^2*dy}}} or {{{pi*(y^(2/3)+2y^(1/3)+1)*dy}}}.&lt;br&gt;
Doing the integration, the volume of my solid of revolution is then pi times {{{(3/5)y^(5/3)+(3/2)y^(4/3)+y}}} evaluated between 0 and 8.&lt;br&gt;
For y=8: {{{(3/5)*32+(3/2)*16 + 8 = 256/5}}}
and of course 0 for y=0.&lt;br&gt;
The volume of my cylinder is {{{pi(3^2)(8) = 72*pi}}}&lt;br&gt;
So the volume we are looking for in the problem is {{{72*pi - (256/5)*pi = (104/5)*pi}}}&lt;br&gt;
... which is the answer the other tutor got.</PRE>