Consider a twice differentiable function $y ( x )$ in an $x y$ plane which connects two points $A ( - 1,2 )$ and $B ( 1,2 )$. Let $S$ be outer surface area of the cylindrical object created by rotation of the curve $y ( x )$ about the $x$ axis. Answer the following questions.
(1) Prove that the surface area $S$ is given by
$$\begin{aligned} S & = 2 \pi \int _ { - 1 } ^ { 1 } F \left( y , y ^ { \prime } \right) \mathrm { d } x \\ F \left( y , y ^ { \prime } \right) & = y \sqrt { 1 + \left( y ^ { \prime } \right) ^ { 2 } } \end{aligned}$$
where $y ^ { \prime } = \frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2) Let the curve $y ( x )$ satisfy the following Euler-Lagrange equation for arbitrary $x$:
$$\frac { \partial F } { \partial y } - \frac { \mathrm { d } } { \mathrm {~d} x } \frac { \partial F } { \partial y ^ { \prime } } = 0$$
Considering Eq. (2.3) along with $\frac { \mathrm { d} F } { \mathrm {~d} x }$, prove that the following relation holds:
$$F - y ^ { \prime } \frac { \partial F } { \partial y ^ { \prime } } = c$$
Here $c$ is a constant.
(3) Express a differential equation satisfied by the curve $y ( x )$ using $y , y ^ { \prime } , c$.
(4) Represent the curve $y ( x )$ as a function of $x$ and $c$.
Obtain an equation which should be satisfied by the constant $c$.