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A Parametric description with respect to arc length, curvature. D Find ds and then compute. I 1) As s (t) = r (t) and r (t) = a sin t2 , cos t2 , f˚ as s (t) = a sin2 (t2 ) + cos2 (t2 ) = a, we get s(t) = at and t(s) = 1 s. a The parametric description with the arc length is x = r(t) = a t 0 sin u2 du , t 0 cos u2 du =a s a 0 sin u2 du , s a 0 cos u2 du . 5 –1 Figure 44: The clothoid for a = 1 and s ∈ [−4, 4]. 2) From r (t) = a sin t2 , cos t2 ∼ a sin t2 , cos t2 , 0 , and r (t) = 2ta cos t2 , − sin t2 ∼ 2ta cos t2 , − sin t2 , 0 , follows that 2ta cos t2 −2ta sin t2 0 0 0 1 {ez × r (t)} · r (t) = a sin t 2 a cos t 2 =− 2ta cos t2 −2ta sin t2 a sin t2 a cos t2 0 = −2ta2 .

A Parametric description with respect to arc length, curvature. D Find ds and then compute. I 1) As s (t) = r (t) and r (t) = a sin t2 , cos t2 , f˚ as s (t) = a sin2 (t2 ) + cos2 (t2 ) = a, we get s(t) = at and t(s) = 1 s. a The parametric description with the arc length is x = r(t) = a t 0 sin u2 du , t 0 cos u2 du =a s a 0 sin u2 du , s a 0 cos u2 du . 5 –1 Figure 44: The clothoid for a = 1 and s ∈ [−4, 4]. 2) From r (t) = a sin t2 , cos t2 ∼ a sin t2 , cos t2 , 0 , and r (t) = 2ta cos t2 , − sin t2 ∼ 2ta cos t2 , − sin t2 , 0 , follows that 2ta cos t2 −2ta sin t2 0 0 0 1 {ez × r (t)} · r (t) = a sin t 2 a cos t 2 =− 2ta cos t2 −2ta sin t2 a sin t2 a cos t2 0 = −2ta2 .

1 + Y (x)2 and reduce the line integral of D Sketch the plane curve. Calculate the weight function integrand 1 to an ordinary integral. 1. com 32 Calculus 2c-7 Arc lengths and parametric descriptions by the arc length I 1) It follows from 2 x3 + , 24 x Y (x) = thus Y (x) = 2 x2 x4 − 16 − 2 = , 8 x 8x2 that 1 + Y (x)2 = 1 8x2 64x4 + (x4 − 16)2 = 2 1 x2 + 2. (x4 + 16) = 2 8x 8 x We get the arc length by insertion, 4 K 4 x3 2 2 x2 + 2 dx = − 8 x 24 x 2 2 56 1 7 1 17 64 − 8 2 2 − + = − +1= + = . 2 for a = 1.

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