Question

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 =

Answer 1

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\|(\mathrm d\mathbf r(t))/(\mathrm dt)\right\|\,\mathrm dt`

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

`\mathrm ds=√(1+t^2)\,\mathrm dt`

So we have

`S=\displaystyle\int_0^(3\pi)√(1+t^2)\,\mathrm dt\approx46.132`