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

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<b):
My book procures the formula
s=integral from a to b of 2*pi*y(x)*sqrt(1+(dy/dx)^2)*dx
While I understand how this formula is attained and why, when I think about it, I find no reason why the following formula shouldn't be equally valid:
s=integral from a to b of 2*pi*y(x)*dx

anonymous · Answer

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).