volume of revolution - torus Watch

Hasufel
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#1
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hey, guys!

i`ve been revising these questions, and there`s one that i`ve never done before which has stumped me:

"find the volume of the torus generated by revolving the circle:

(x-a)^{2}+y^{2}=b^{2} about the y axis, where 0<a<b."

i`ve tried to tackle this like, revolving just the part in the 1st quadrant, then quadrupling it, but i got the integrand incorrect.

any hints?
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TeeEm
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(Original post by Hasufel)
hey, guys!

i`ve been revising these questions, and there`s one that i`ve never done before which has stumped me:

"find the volume of the torus generated by revolving the circle:

(x-a)^{2}+y^{2}=b^{2} about the y axis, where 0<a<b."

i`ve tried to tackle this like, revolving just the part in the 1st quadrant, then quadrupling it, but i got the integrand incorrect.

any hints?
You need to set up the integral by the "shell" method
Look at a standard "American Hardback" called calculus ...

the method is not hard but very hard to explain on line.

OR look at U tube.

By the way I will do your problem (to add to my resources) but I won't have time until Thursday). PM me on Thursday evening if you haven't done it by then.
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DFranklin
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(Original post by Hasufel)
hey, guys!

i`ve been revising these questions, and there`s one that i`ve never done before which has stumped me:

"find the volume of the torus generated by revolving the circle:

(x-a)^{2}+y^{2}=b^{2} about the y axis, where 0<a<b."

i`ve tried to tackle this like, revolving just the part in the 1st quadrant, then quadrupling it, but i got the integrand incorrect.
I think the easiest way of doing this is to divide into cylinderical "shells" (i.e. integrate over the radius (sqrt(x^2+y^2)), as it goes from the inner radius to the outer radius of the torus.

Second easiest is going to be to integrate over y, and directly calculate cross-sectional area of the torus as a function of y.
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TeeEm
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(Original post by Hasufel)
hey, guys!

i`ve been revising these questions, and there`s one that i`ve never done before which has stumped me:

"find the volume of the torus generated by revolving the circle:

(x-a)^{2}+y^{2}=b^{2} about the y axis, where 0<a<b."

i`ve tried to tackle this like, revolving just the part in the 1st quadrant, then quadrupling it, but i got the integrand incorrect.

any hints?

(Original post by TeeEm)
You need to set up the integral by the "shell" method
Look at a standard "American Hardback" called calculus ...

the method is not hard but very hard to explain on line.

OR look at U tube.

By the way I will do your problem (to add to my resources) but I won't have time until Thursday). PM me on Thursday evening if you haven't done it by then.
I have done the integral in Cartesian from first principles. If you have no satisfactory solution PM me
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Hasufel
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(Original post by TeeEm)
I have done the integral in Cartesian from first principles. If you have no satisfactory solution PM me
thanks - will rep when can (this thing won`t let me) - i`ve found 3 ways (D Franklin`s Pappus` 2nd theorem) and 2 others - will look at when get chance -

moved on now to Complex Analysis and Cauchy-Goursat.

Thanks again!

(this was just one problem i was curious about - i was never taught this at university! can you believe that? )
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TeeEm
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(Original post by Hasufel)
thanks - will rep when can (this thing won`t let me) - i`ve found 3 ways (D Franklin`s Pappus` 2nd theorem) and 2 others - will look at when get chance -

moved on now to Complex Analysis and Cauchy-Goursat.

Thanks again!

(this was just one problem i was curious about - i was never taught this at university! can you believe that? )
It is all good.

Firstly Pappus is good but this is just a formula unless you prove it first.

Secondly not all things can be taught at university, so nobody to blame.

Thirdly your enquiry made me add the proof in my own resources which I will put on my website in my next update, so thank you for that.

goodnight
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DFranklin
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(Original post by Hasufel)
(this was just one problem i was curious about - i was never taught this at university! can you believe that? )
I was never taught this either. But once solved an S-level question to find the Moment of Inertia of a torus and it's kind of stuck with me. (Note that if you go the radial route this is hardly more difficult than finding the volume).
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ghostwalker
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(Original post by Hasufel)
...
(Original post by DFranklin)
...
Now that it's been mentioned, I can just about recall Pappus' theorems from A-level. T'was in Plumpton and Tomkys Sixth Form Pure Mathematics.
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