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.
2026-03-30 23:13:14.1774912394
Volumes of solids formed by revolution of curve
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1
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.