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Extra resources for Calculus 2c-7, Examples of Line Integrates

Example text

D Find the arc length s, and then ψ by a geometrical analysis. 2. 1) The point P0 has the coordinates (0, a). g. r(t) = t, a cosh t a , hence r (t) = 1, sinh t a where r (0) = (1, 0), , and thus ψ = Arctan sinh t . a From s (t) = r (t) = 1 + sinh2 t t = cosh , a a follows that t s(t) = cosh 0 u u dy = a sinh a a t 0 = a sinh t a . The natural equation is ψ = Ψ(s) = Arctan s . a 2) The point P0 has the coordinates (−a, 0), and r (t) = a 3 cos2 t · sin t, 3 sin2 t · cos t = 3a cos t · sin t(cos t, sin t).

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.

Then compute the tangential line integral. 1. com 57 Calculus 2c-7 Tangential line integrals I 1) Here the parametric description of the curve can also be written y= x 2−x for x ∈ [0, 1], for x ∈ [1, 2]. Please click the advert it’s an interesting world Where it’s Student and Graduate opportunities in IT, Internet & Engineering Cheltenham | £competitive + benefits Part of the UK’s intelligence services, our role is to counter threats that compromise national and global security. We work in one of the most diverse, creative and technically challenging IT cultures.

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