Engineering Calculus Notes 213

Engineering Calculus Notes 213 - 201 2.5. INTEGRATION ALONG...

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Unformatted text preview: 201 2.5. INTEGRATION ALONG CURVES differential of arclength, denoted ds, to be the formal expression → ˙ p ds := − (t) dt = x(t)2 + y (t)2 + z (t)2 dt. ˙ ˙ ˙ This may seem a bit mysterious at first, but we will find it very useful; using this notation, the content of Theorem 2.5.1 can be written b s (C ) = ds. a As an example, let us use this formalism to find the length of the helix parametrized by x(t) = cos 2πt y (t) = sin 2πt z ( t) = t or → − (t) = (cos 2πt, sin 2πt, t) p 0≤t≤2: we have x(t) = −2π sin 2πt ˙ y (t) = 2π cos 2πt ˙ z (t) = 1 ˙ so (x(t))2 + (y (t))2 + (z (t))2 dt ˙ ˙ ˙ ds = = (−2π sin 2πt)2 + (2π cos 2πt)2 +)(1)2 dt = 4π 2 + 1 dt and 2 s (C ) = ds 0 2 4π 2 + 1 dt = 0 = 2 4π 2 + 1. ...
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