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Lesson 14 - Length of an Arc - = = LENGTH OF AN...

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TOPIC LENGTH OF AN ARC
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OBJECTIVES discuss the formula for computing length of an arc; compute the length of an arc.
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Length of curves To find the length of the arc of the curve y = f ( x ) between x = a and x = b let δ s be the length of a small element of arc so that: 2 2 2 2 ( ) ( ) ( ) so 1 s x y s y x x δ δ δ δ δ δ δ 2245 + 2245 + ÷
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Length of curves In the limit as the arc length δ s approaches zero: and so: 2 1 ds dy dx dx = + ÷ 2 1 b x a b x a ds s dx dx dy dx dx = = = = + ÷
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Length of curves – parametric equations Instead of changing the variable of the integral as before when the curve is defined in terms of parametric equations, a special form of the result can be established which saves a deal of working when it is used. Let: 2 1 2 2 2 2 2 2 2 2 2 2 ( ) and ( ). As before ( ) ( ) ( ) so so as 0 and t t t y f t x F t s x y s x y t t t t ds dx dy dx dy s dt dt dt dt dt dt δ δ δ δ δ δ δ δ δ δ = = = 2245 + 2245 + ÷ ÷ ÷ = + = + ÷ ÷ ÷ ÷
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Unformatted text preview: = + = + ÷ ÷ ÷ ÷ ∫ LENGTH OF AN ARC A. Rectangular Coordinates f(x) y , 1 2 = + = dx dx dy ds g(y) x , dy dy dx 1 ds 2 = + = B. Parametric Form dt dt dy dt dx ds 2 2 + = when x=x(t), y=y(t); where t is a parameter 1. 2. C. Polar Coordinates f(r) , dr dr d r 1 ds 2 2 = θ θ + = ) g( r , d d dr r ds 2 2 θ = θ θ + = 2. 1. EXAMPLE Find the length of the arc of each of the following: 2 3 3 3 t y t t x =-= 1. from t = 0 to t = 1 2. t e y t e x t t sin cos = = from t = 0 to t = 4 4. Length of the arc of the semicircle 2 2 2 a y x = + 3. HW #3 Find the exact arc length of the curve over the interval....
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