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Section 12.9 : Arc Length with Vector Functions

For problems 1 – 3 determine the length of the vector function on the given interval.

1. \(\vec r\left( t \right) = 4\cos \left( {2t} \right)\vec i + 3t\,\vec j - 4\sin \left( {2t} \right)\vec k\) from \(0 \le t \le 3\pi \).
2. \(\vec r\left( t \right) = \left\langle {9 - 2t,4 + 2t,\sqrt 2 \,{t^2}} \right\rangle \) from \(0 \le t \le 1\).
3. \(\vec r\left( t \right) = 2t\,\vec i + \frac{1}{2}{t^2}\,\vec j + \ln \left( {{t^2}} \right)\vec k\) from \(1 \le t \le 3\).

For problems 4 – 6 find the arc length function for the given vector function.

1. \(\vec r\left( t \right) = \left\langle {8t,6 + t, - 7t} \right\rangle \)
2. \(\displaystyle \vec r\left( t \right) = \left\langle {8t,4{t^{\frac{3}{2}}},3} \right\rangle \)
3. \(\vec r\left( t \right) = \left\langle {{{\bf{e}}^{4t}}\sin \left( t \right),{{\bf{e}}^{4t}}\cos \left( t \right),2} \right\rangle \)
4. Determine where on the curve given by \(\vec r\left( t \right) = \left\langle {8t,4{t^{\frac{3}{2}}},3} \right\rangle \) we are after traveling a distance of 4.
5. Determine where on the curve given by \(\vec r\left( t \right) = \left\langle {{{\bf{e}}^{4t}}\sin \left( t \right),{{\bf{e}}^{4t}}\cos \left( t \right),2} \right\rangle \) we are after traveling a distance of 15.