(t = < f(t g(t h(t > L = |t=bt=a(f(t)2 g(t)2 h(t)2)1/2 dt = |t=bt=a ">
13-3 Area Length and currature

# 13-3 Area Length and currature - 13.3 Area Length and...

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13.3 Area Length and currature "K" (kappa) Def r -> (t) = < f(t) , g(t), h (t) > L = | t=b t=a (f(t) 2 + g(t) 2 h(t) 2 ) 1/2 dt = | t=b t=a ||r -> '(t)|| dt Ex: Find the arc-length r -> (t) = <t 2 , 2t, lnt> where 1<=t<= e> r -> (t) = <2t, 2, 1/t> ||r -> '(t)|| = ((2t) 2 +2 2 +(1/t) 2 ) 1/2 = (4t 2 + 4 + 1/t 2 ) 1/2 = ((4t 4 +4t 2 +1)/t 2 ) 1/2 = (4t 4 +4t 2 +1) 1/2 /(t 2 ) 1/2 = ((2t+1) 2 ) 1/2 /t = (2t+1)/t L = | t=e t=1 (2t 2 + 1)/t dt = | e 1 (2t + 1/t) dt = t 2 + ln(t) | e 1 = e 2 units of length currature K K(t) = ||dT -> /ds|| rate of change of the unit tangent, vector with respect to the arc- length For computational purposes(when we do our hw): K(t) = ||r -> ' (t) x r -> ''(t)|| / ||r -> ' (t)|| 3 Ex: Find the curvature of r -> (t) = <t 2 ,2t,lnt> @ P(1,2,0) r -> '(t) = <2t,2,1/t> r -> "(t) = <2, 0 , -1/t 2 > r -> (t) x r -> "(t) = |i -> j -> k -> | 2t 2 1/t |2 0 1/t | K(t) = (r/t 2 + 16/t 2 + 16) 1/2 /(4t 2 +4t+1/t 2 ) 3 | t=1 = 2/9 - Curvature K measures the amount of "bend" in the curve. Note: The curvature of a line is zero.

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Note: The curvature of a circle centered @ (0,0), radius "a" : 1/a = K If y = f(x) K(x) = | f"(x)|/ [1+(f '(x)) 2 ] 3/2 Ex: For y = x 2 f '(x) = 2x f "(x) = 2 K(2) = ? = |2|/[1+(2x) 2 ] 3/2 unit tangent vector: T
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