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Arc Length and Surface Area. Lesson 10.8. The Sequel. Using Parametric Equations. Recall formula for arc length If x = f(t) and y = g(t) it can be shown that. Example. Given x = sin t, y = cos t What is the arc length from t = 0 to t = 2 π Determine dx/dt and dy/dt

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arc length and surface area

Arc Length and Surface Area

Lesson 10.8

The Sequel

using parametric equations
Using Parametric Equations
  • Recall formula for arc length
  • If x = f(t) and y = g(t) it can be shown that
example
Example
  • Given x = sin t, y = cos t
    • What is the arc length from t = 0 to t = 2π
  • Determine dx/dt and dy/dt
    • dx/dt = cos t dy/dt = -sin t
  • Now what is the integral?
using polar equations
Using Polar Equations
  • Given a curve in polar form r = f (θ)
    • Must have continuous first derivative on interval
    • Curve must be traced exactly once for a ≤ θ ≤ b
  • Arc length is
try it out
Try it Out!
  • Given polar function
    • What is the arc length from θ = 0 to θ = 4
  • Find dr/dθ
  • What is the integral and its evaluation
surface area parametric form

Change this to x if revolved about y-axis

Surface Area – Parametric Form
  • Recall formula for surface area of rectangular function revolved about x-axis
  • Formula for parametric form about x-axis
surface area example
Surface Area Example
  • Given x = t, y = 4 – t2 from t = 0 to t = 2
    • Surface area if revolved around x-axis
surface area polar form
Surface Area – Polar Form
  • Curve revolved around x-axis
  • Curve revolved around y-axis
find that surface area
Find That Surface Area
  • Given r = sin θ, θ = 0 to θ = π/2
    • Revolve about polar (x) -axis
assignment
Assignment
  • Lesson 10.8
  • Page 451
  • Exercises 1 – 21 odd