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202303141137 Exercise 2.7.1

Calculate the lengths of the following smooth simple arcs in \mathbb{R}^3.

(a) The circular helix parametrized by

f(t)=(t,\cos t,\sin t)

where t\in [a,b];
(b) the curve parametrized by

f(t)=(e^t\cos t,e^t\sin t,e^t)

where t\in [0,k].

Extracted from P. R. Baxandall. (1986). Vector Calculus.

Roughwork.

Let f:[a,b]\subseteq\mathbb{R}\to\mathbb{R}^n be a C^1 path in \mathbb{R}^n. The length of f is defined to be

l(f)=\displaystyle{\int_{a}^{b}\|f'(t)\|\,\mathrm{d}t}.

See Definition 2.7.2 on pg.60

(a)

\begin{aligned} f(t) & = (t,\cos t,\sin t) \\ f'(t) & = (1,-\sin t,\cos t) \\ \|f'(t)\| & = \sqrt{(1)^2+(-\sin t)^2+(\cos t)^2} \\ & = \sqrt{2} \\ l(f) & = \int_{a}^{b} \sqrt{2}\,\mathrm{d}t \\ & = \sqrt{2}(b-a) \\ \end{aligned}

(b) Not to be attempted.