Arc Length and Surface Area

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# Arc Length and Surface Area - PowerPoint PPT Presentation

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

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