Arc lengths and surface areas

Arc lengths and surface areas

Arc Lengths

  • The arc length is the length of a curve between two points.
  • For a curve defined by a function y = f(x) from x = a to x = b, the formula for the arc length is: L = ∫(from a to b) √(1+(dy/dx)^2) dx, where dy/dx is the derivative of the function.
  • In parametric form, where the curve is defined by two functions x = g(t) and y = h(t), from t = a to t = b, the arc length is calculated using: L = ∫(from a to b) √((dx/dt)^2 + (dy/dt)^2) dt.
  • When dealing with polar coordinates, where a curve is defined by a function r = r(θ) from θ = α to θ = β, the arc length becomes: L = ∫(from α to β) √(r^2 +(dr/dθ)^2) dθ.
  • Calculating arc lengths often involves complex integration, and numerical methods may be required if the integral cannot be easily evaluated.

Surface Areas of Revolution

  • The surface area of a solid figure generated by revolving a curve around an axis can also be calculated.
  • If y = f(x), a ≤ x ≤ b, revolves around the x-axis, the surface area, A, is given by: A = 2π∫(from a to b) y√(1+(dy/dx)^2) dx.
  • For a curve defined parametrically by x = g(t), y = h(t), a ≤ t ≤ b, revolved around the x-axis, the formula alters to: A = 2π∫(from a to b) y√((dx/dt)^2 + (dy/dt)^2) dt.
  • In polar coordinates, for a curve r = r(θ), α ≤ θ ≤ β, revolved around the polar axis, the surface area is: A = 2π∫(from α to β) r sin(θ)√(r^2 +(dr/dθ)^2) dθ.
  • Note: make sure to use the correct formula depending on the method of defining the curve (function, parametric or polar) and which axis the shape is being revolved around.

Additional Points for Integration

  • Understanding and general proficiency in integration is an absolute necessity for arc length and surface area applications.
  • Sometimes, integrals can’t be simplified analytically and either approximation methods or numerical evaluations with technology might be necessary.
  • Be comfortable using trigonometric identities or substitution methods to simplify complex integrals.
  • Implicit differentiation might be required in some cases when dealing with parametric or polar functions.