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 Determine dx/dt and dy/dt 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 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! Given polar function Find dr/dθ 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 Recall formula for surface area of rectangular function revolved about x-axis Formula for parametric form about x-axis Change this to x if revolved about y-axis

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 Curve revolved around x-axis Curve revolved around y-axis

Find That Surface Area Given r = sin θ, θ = 0 to θ = π/2 Revolve about polar (x) -axis

Assignment Lesson 10.8 Page 451 Exercises 1 – 21 odd