If I try to find the surface area of any solid by using cylindrical slices, I'm getting wrong answer. I'm taking this as the formula,
$$S = \int_a^b2\pi y dx$$
where $y$ = height ($2\pi y$ = circumference of the cylinder) $dx$ = width. We can approximate the surface area using cylindrical shells right? If we can approximate volume, we can also approximate surface area right? This is my argument... Please help.
Actually, approximating surface area by cylindrical shells doesn't work, for the same reason that $\pi \neq 4$ in this thread http://www.physicsforums.com/showthread.php?t=452917
Cylindrical shells do not give the correct "small" surface element because they are all "almost" parallel to the axis of revolution.
The correct formula for $y=f(x)$, $a \leq x \leq b$ to find the surface area of the surface formed by revolving $f$ around the $x$-axis is $$ S=2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}dx. $$
More information on this topic can be found at http://en.wikipedia.org/wiki/Surface_of_revolution or by googling "surface area by revolution".
