Suppose $y=f(x)$ is a continuous curve on $[a,b]$.Suppose we are to find the volume of revolution of the solid generated by the area under $f(x)$ and bounded by $x$-axis and the ordinates $x=a$ and $x=b$.
We consider the solid to be composed of elementary cylindrical slices of width $\delta x$ and radius $f(x)$ and we integrate their volumes to get the formula as $\int_a^b \pi [f(x)]^2 dx$.
Note that while we are considering cylindrical slices we are neglecting the curvature of $f(x)$ and the logic is roughly given as $\delta x$ is very small,so the difference between $f(x)$ and $f(x+\delta x)$ is small by continuity and so we can consider it as $f(x)$ only and neglect the curvature.
The logic seems fine.But when we come to calculating the surface of revolution of the surface generated by rotating $f(x)$ curve about $x$-axis,we give some different logic.
We do not consider cylindrical rings of thickness $\delta x$ in this case.Rather we take the curvature of $f(x)$ into account and consider a bend ring as shown in the second diagram below?If we take $\delta x$ thickness and consider cylindrical rings of radius $f(x)$ then I am getting wrong answer.
Why is it so that for volume and area we have different explanations?