surface Areas using cylindrical shells
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.
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$\begingroup$Actually, approximating surface area by cylindrical shells doesn't work, for the same reason that $\pi \neq 4$ in this thread
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 or by googling "surface area by revolution".
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