Mathematics: Find the length S of the spiral (t cos(t), t sin(t)) for 0 ≤ t ≤ 3π. (Round your answer to three decimal places.) S =

exercise educationExplanation:...

Find the length S of the spiral (t cos(t), t

ccollinsphotogr3177

30.01.2022

The arc length is

$S=\displaystyle\int_C\mathrm ds$

where C is the given curve and ds is the line element. C is defined on 0 ≤ t ≤ 3π by the vector function,

$\mathbf r(t)=(t\cos t,t\sin t)$

so the line element is

$\mathrm ds=\left\|\dfrac{\mathrm d\mathbf r(t)}{\mathrm dt}\right\|\,\mathrm dt$

$\mathrm ds=\sqrt{\left(\dfrac{\mathrm d(t\cos t)}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm d(t\sin t)}{\mathrm dt}\right)^2}\,\mathrm dt$

$\mathrm ds=\sqrt{1+t^2}\,\mathrm dt$

So we have

$S=\displaystyle\int_0^{3\pi}\sqrt{1+t^2}\,\mathrm dt\approx46.132$

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