Question

Hi, I'm having a hard time understanding the difference in validity between two formulas for the surface area of a solid produced by rotation around the x axis between a and b (a 11 years ago

Answer 1

The two formulas are only equivalent if \(y(x)\) is a constant function.
\[A=2\pi\int_Cy~dS=2\pi\int_a^by(x)\sqrt{1+\left[\frac{dy}{dx}\right]^2}~dx\]
Consider the functions \(y=1\) and \(y=x\) over the intervals \([0,1]\). The region of interest is to be revolved about the \(x\)-axis.
\[A_1=2\pi\int_0^1\sqrt{1}~dx=2\pi\]
(Note that this is the lateral surface area of a cylinder of radius and height 1).
\[A_2=2\pi\int_0^1x\sqrt{1+1}~dx=\sqrt2\pi\]
(Note that this is the lateral surface area of a cone of radius and height 1).