Arc Length and Surface Area

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 = 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 • What is the arc length from θ = 0 to θ = 4 • Find dr/dθ • What is the integral and its evaluation

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 • 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

Arc Length and Surface Area