12.9-bo'lim : Vektor funktsiyalari bilan yoy uzunligi

1 – 3 masalalar uchun vektor funksiyaning berilgan oraliqdagi uzunligini aniqlang.

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

4-6-masalalar uchun berilgan vektor funksiya uchun yoy uzunligi funksiyasini toping.

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. 4 masofani bosib o‘tganimizdan keyin \(\vec r\left( t \right) = \left\langle {8t,4{t^{\frac{3}{2}}},3} \right\rangle \) tomonidan berilgan egri chiziqning qayerida ekanligimizni aniqlang.
5. 15 masofani bosib o‘tganimizdan keyin \(\vec r\left( t \right) = \left\langle {{{\bf{e}}^{4t}}\sin \left( t \right),{{\bf{e}}^{4t}}\cos \left( t \right),2} \right\rangle \) tomonidan berilgan egri chiziqning qayerida ekanligimizni aniqlang.