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Additional info for Calculus 2c-7, Examples of Line Integrates

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7 A space curve K is given by the parametric description √ r(t) = e3t , e−3t , 18 t , t ∈ [−1, 1]. Prove that r (t) = 3 e3t + 3−3t , and ﬁnd ﬁnd the arc length of K. A Arc length. D Find r (t). com 52 Calculus 2c-7 Arc lengths and parametric descriptions by the arc length 4 2 0 5 –2 5 –4 10 10 15 15 20 20 Figure 46: The curve K. and we get the arc length 1 (K) r (t) dt = = −1 1 = 2 Please click the advert 0 1 −1 3 e3t + e−3t dt 3 · 2 cosh 3t dt = 4[sinh 3t]10 = 4 sinh 3. We have ambitions. Also for you.

6. Furthermore, the curves of these two examples have the same initial point and end point. Nevertheless the two tangential line integrals give diﬀerent results. We shall later be interested in those vector ﬁelds V(x), for which the tangential line integral only depends on the initial and end points of the curve K. ) We have here an example in which this ideal property is not satisﬁed. 7. 7) We get K V(x) · dx = π = π 2 π = π 2 {sin4 t + cos t + 3 cos2 t + 3 cos3 t + cos4 t}dt sin4 t + cos4 t + 2 cos2 t · sin2 t − π 2 π = (sin2 t + cos2 t)2 − π 2 = t− = {−y 3 dx + x3 dy} {− sin3 t · (− sin t) + (1 + cos t)3 cos t}dt π = K 1 3 3 sin2 2t + cos t + + cos 2t + 3 cos3 y dt 2 2 2 3 1 1 + cos 4t + cos t + cos 2t + 3 cos t − 3 sin2 t cos t dt 4 4 2 1 3 3 t + sin 4t + sin t + t + sin 2t + 3 sin t − sin3 t 4 16 2 4 1− 1 3 + 4 2 π π 2 9π π −4+1= − 3.

A Arc length. D Find r (t) . 5 1 Figure 47: The curve K. I It follows from r (t) = (t2 − 1, t2 + 1, 2t), that r (t) 2 = (t2 −1)2 +(t2 +1)2 +4t2 = 2t4 +2+ 4t2 = 2(t2 + 1)2 , hence 1 (K) = −1 r (t) dt = 2 1 √ 0 2 √ 2(t + 1) dt = 2 2 1 +1 3 √ 8 2 = . 9 A space curve K is given by the parametric description √ r(t) = 6t2 , 4 2 t3 , 3t4 , t ∈ [−1, 1]. Explain why the curve is symmetric with respect to the (X, Z)-plane. Then ﬁnd the arc length of K. A Arc length. D Replace t by −t. Then ﬁnd r (t). com 54 Calculus 2c-7 Arc lengths and parametric descriptions by the arc length –4 3 2 –2 1 0 2 1 2 4 3 4 5 6 Figure 48: The curve K.