2.4.4—Arc length and surface area

For y=f(x), arc length from a to b is ∫√(1+(dy/dx)²)dx. Rotating the curve about an axis gives surface area 2π∫(radius)×(arc-length element), with the radius and variable chosen consistently.

Parametric and polar curves require their own ds formula. Find endpoints and avoid counting a self-intersecting curve or using signed radius without considering the geometry.

For y=x² from 0 to 1, ds=√(1+4x²)dx; surface area about the x-axis uses radius y=x², so the integrand is 2πx²√(1+4x²).

Arc length is not ∫dy or ∫|f|dx, and surface area is not volume; use the local slant factor.