A complete revolution is described around a X axis by a segment of curve C between the two points $P\left(x_1, y_1\right)$ and $P^{\prime}\left(x_2, y_2\right)$ such that the obtained revolution surface $S$ is minimal. (1) Show that the surface S is expressed by

$$ S=2 \pi \int_{x_1}^{x_2} y \sqrt{1+y^{\prime 2}} d x $$

(2) Show that the differential equation of the curve $C$ is

$$ y y^{\prime \prime}=1+y^{\prime 2} $$

(3) Determine the shape of the curve $C$ so that the surface $S$ is minimal.

🧠Catenoid Surface

https://gist.github.com/viadean/6857283dc63fdd9649a052d987159f2c

🧠Analysis in theory

1. Surface Area of Revolution

• Understanding the Setup:
  • We have a curve C in the xy-plane defined by y = f(x) between points P(x₁, y₁) and P'(x₂, y₂).
  • This curve is rotated around the x-axis, creating a surface of revolution S.
• Derivation of the Surface Area:
  • Consider a small segment of the curve C with length ds.
  • When this segment is rotated around the x-axis, it forms a ring with radius y and width ds.
  • The area of this ring is 2πy ds.
  • The total surface area S is the integral of these ring areas: S = ∫ 2πy ds.
  • We know that ds = √(dx² + dy²) = √(1 + (dy/dx)²) dx = √(1 + y'²) dx.
  • Therefore, the surface area S is given by:
    • S = 2π ∫[x₁, x₂] y √(1 + y'²) dx

2. Differential Equation of the Curve

• Variational Calculus:
  • To find the curve that minimizes the surface area S, we need to use variational calculus.
  • We want to minimize the integral: ∫[x₁, x₂] y √(1 + y'²) dx.
  • Let F(y, y') = y √(1 + y'²).
  • The Euler-Lagrange equation is: ∂F/∂y - d/dx (∂F/∂y') = 0.
• Calculating Partial Derivatives:
  • ∂F/∂y = √(1 + y'²)
  • ∂F/∂y' = y y' / √(1 + y'²)
• Applying Euler-Lagrange Equation:
  • √(1 + y'²) - d/dx (y y' / √(1 + y'²)) = 0
  • √(1 + y'²) = d/dx (y y' / √(1 + y'²))
  • (1 + y'²) = d/dx (y y') / √(1 + y'²) * √(1+y'²)
  • (1+y'²) = d/dx(yy')
  • (1 + y'²) = y'y' + yy''
  • yy'' = 1 + y'²
• Result:
  • The differential equation of the curve C is yy'' = 1 + y'².

3. Shape of the Curve

• Solving the Differential Equation:
  • The differential equation yy'' = 1 + y'² is a second-order nonlinear differential equation.
  • Let y' = p, so y'' = dp/dx = (dp/dy)(dy/dx) = p dp/dy.
  • The equation becomes: yp dp/dy = 1 + p².
  • Separate variables: y dy = p dp / (1 + p²).
  • Integrate both sides: ∫ y dy = ∫ p dp / (1 + p²).
  • y²/2 = (1/2) ln(1 + p²) + C₁.
  • y² = ln(1 + p²) + 2C₁.
  • Let 2C₁ = ln(C₂²), where C₂ is a constant.
  • y² = ln(1 + p²) + ln(C₂²) = ln(C₂²(1 + p²)).
  • e^(y²) = C₂²(1 + p²).
  • p² = (e^(y²)/C₂²) - 1.
  • p = dy/dx = ±√((e^(y²)/C₂²) - 1).
  • This leads to the equation of a catenary.
• Catenary:
  • The solution to this differential equation is a catenary, which is the curve formed by a hanging chain or cable under its own weight.
  • The general form of the catenary is: y = a cosh(x/a), where a is a constant.
  • In our case, the catenary is rotated around the x-axis, forming a catenoid.
• Conclusion:
  • The curve C that minimizes the surface area of revolution is a catenary. The surface S is a catenoid.