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While finding the volumes of solids formed by revolution of 2-D curve , I think we should consider frustum shaped elemental rigs rather than cylindrical because in cylindrical part volume calculated may be more than the actual one. Also, when we are a curve and we consider cylindrical elemental rings than it indirectly indicates that the slope of function at every point is 0. So as per above discussion I think that we should consider frustum shaped rings ? But in some places cylindrical rings are also used ? I am really confused what to do ? Please resolve my doubt . I am just a beginner in maths.

ronak jain
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  • Both approaches (and others) will work. With a concrete example why not try to get the same answer via the two methods ? – Martin Hansen Jul 11 '20 at 15:18

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What you say concerns surface area more than volumes because slope $y'$ is not involved in the integrand for volume function definition for differential rings.

$$ S.A. = \int 2 \pi y \sqrt{1+y^{'2}} dx$$

$$ Volume = \int \pi y ^2 dx$$

There is no harm considering frustum of cones but it is wasted effort due to disconnect of volume with slope. Elemental rings summation is sufficient here.

Narasimham
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  • That's fine but when I try to find some expression in physics like electric potential due to sphere using Cartesian coordinate not polar. Then answer doesn't comes out to be true. Some experts say to use polar coordinates or to consider elemental ring as frustum . And than answer comes true. And at that point I got confused. Please explain. – ronak jain Jul 11 '20 at 17:04
  • Please give specific example of the integrand. Result does not depend on the choice of coordinate system. it is so that one is more convenient than the other from its geometry. – Narasimham Jul 11 '20 at 17:09
  • OK. But first please tell me that suppose we wish to find the surface area of sphere than at a angular position "theta" there would beelemental rings which wouldbein shape of frustum than why don't we find elemental area according to frustum . Wejust consider it as cylinder. Though we know that there is difference in area of both shapes. And we may not get exact answer. – ronak jain Jul 11 '20 at 17:14
  • My answer has addressed this question. The definition.. whether involves differential or not makes all the difference. – Narasimham Jul 11 '20 at 17:20