[Blindmath] volume of rotational solids (calculus)

[Alex Hall]

Yes, calc 2, and yes, only continuous functions. The problem is that,
sometimes, there is a region that is undefined for the function. If
you have some exponential, call it e^x, from x=0 to x=2 rotated about
the y-axis, there is a region shaped like a bowl that you must account
for by subtracting it out. For a more complex equation, imagining the
shapes is not so easy, so I could not just know that there is a region
whose area must be subtracted out. I am looking for a non-graphing way
to do this. After all, as you said, sketches are just a visualization
tool; functions should not need to be sketched to be worked with.

On 3/8/11, Salisbury, Justin Mark <SALISBURYJ08 at students.ecu.edu> wrote:
> Hi Alex,
>
>     Are you taking Calculus II?  If so, I'm pretty sure that they aren't
> expecting you to integrate any functions that are undefined.  I'm pretty
> sure you'll be integrating continuous functions.  If you're asking how to
> know which bound lies inside the other one, that's generally a matter of a
> distance from the axis of rotation.  I hope that you can mentally
> conceptualize which functions lie inside other ones.  Otherwise, you'll have
> to use a distance formula to the axis of rotation for points on each
> equation.  If you just have the understanding that you're calculating the
> distance between two or more equations and then rotating it to turn area
> into volume, you should have a thorough enough understanding of what you're
> doing.  If you want to contact me off-list for humorous real-life examples,
> feel free.  Sighted mathematicians really only use sketches to help
> themselves understand what they're doing.  Many advanced mathematicians
> never even draw the functions before beginning their computations.  I think
> you'll be just fine as long as you know that you're accomplishing and know
> which functions lie inside the others.
>
> Good luck!
>
> Justin
>
> Justin M. Salisbury
> Undergraduate Student
> The University Honors Program
> East Carolina University
> salisburyj08 at students.ecu.edu
>
> "It is the mark of an educated mind to be able to entertain a thought
> without accepting it."    -Aristotle
> ________________________________________
> From: blindmath-bounces at nfbnet.org [blindmath-bounces at nfbnet.org] on behalf
> of Alex Hall [mehgcap at gmail.com]
> Sent: Tuesday, March 08, 2011 4:30 PM
> To: Blind Math list for those interested in mathematics
> Subject: [Blindmath] volume of rotational solids (calculus)
>
> Hi all,
> This is not a request for help in finding this sort of thing. Rather,
> I am wondering if it can be done purely algebraically so I do not have
> to try to imagine the graph. Example:
>
> Find the volume of the solid formed by rotating the function y=x^2
> around the x-axis from x=0 to x=4.
>
> This one is a pretty simple example, and should be pi*x^5/5, I think.
> This is using the Disk Method, but what happens with the Washer method
> or the Shell method, where you might have space in the solid where the
> function is not defined? Currently, I have to try to imagine the graph
> to "see" the radius to use, any undefined portions, and so on. What I
> am wondering is if anyone has dealt with this and has found any way to
> do it all with algebra or some other non-graphical method. If so,
> please share! Thanks.
>
> --
> Have a great day,
> Alex (msg sent from GMail website)
> mehgcap at gmail.com; http://www.facebook.com/mehgcap
>
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-- 
Have a great day,
Alex (msg sent from GMail website)
mehgcap at gmail.com; http://www.facebook.com/mehgcap