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9.4-bo'lim : Parametrik tenglamalar bilan yoy uzunligi

Ushbu bo'limdagi barcha masalalar uchun javobni aniqlash uchun faqat berilgan parametrik tenglamalardan foydalaning.

1 – 5 masalalar uchun parametrik tenglamalar to‘plami bilan berilgan parametrik egri chiziq uzunligini aniqlang. Ushbu muammolar uchun siz egri chiziq berilgan t diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.

  1. \(x = 3 + 9t\hspace{0.25in}\,\,y = 10 - 15t\hspace{0.25in} - 5 \le t \le 8\)
  2. \(x = 6{\left( {3 + t} \right)^{\frac{3}{2}}}\hspace{0.25in}\,\,y = - 3{t^{\frac{3}{2}}}\hspace{0.25in} - 2 \le t \le 1\)
  3. \(x = 4{t^2} - 3\hspace{0.25in}\,\,y = 3t\hspace{0.25in}0 \le t \le 5\)
  4. \(x = 3 + t\hspace{0.25in}\,\,y = 6 + {\left( {t - 1} \right)^2}\hspace{0.25in}1 \le t \le 3\)
  5. \(x = {t^2} - 1\hspace{0.25in}\,\,y = {t^4} + 5\hspace{0.25in}0 \le t \le 1\)

6 va 7-masalalar uchun zarracha quyidagi parametrik tenglamalar to'plami bilan aniqlangan yo'l bo'ylab harakatlanadi. Zarracha bosib o'tgan umumiy masofani aniqlang va uni parametrik egri chiziq uzunligi bilan solishtiring.

  1. \(\displaystyle x = 6{\cos ^2}\left( {3t} \right)\hspace{0.25in}y = 2 - 3{\sin ^2}\left( {3t} \right)\hspace{0.25in}\hspace{0.25in}\,\,\,\, - \frac{5}{6}\pi \le t \le 3\pi \)
  2. \(\displaystyle x = 3 + \cos \left( {\frac{1}{6}t} \right)\hspace{0.25in}y = 4 + {\cos ^2}\left( {\frac{1}{6}t} \right)\hspace{0.25in}\,\,\,\,\,\,\, - 90\pi \le t \le 216\pi \)

8 - 10 muammolar uchun parametrik tenglamalar to'plami tomonidan berilgan parametrik egri chiziq uzunligini beradigan integralni o'rnating, lekin baholamang. Ushbu muammolar uchun siz egri chiziq berilgan t diapazoni uchun aynan bir marta chiqadi deb taxmin qilishingiz mumkin.

  1. \(x = t\cos \left( {2t} \right)\hspace{0.25in}\,\,y = \sin \left( {3t} \right)\hspace{0.25in}2 \le t \le 3\)
  2. \(x = 1 - \sin \left( {1 + \sqrt t } \right)\hspace{0.25in}\,\,y = \sin \left( {{{\bf{e}}^{ - t}}} \right)\hspace{0.25in}1 \le t \le 4\)
  3. \(\displaystyle x = \ln \left( {t + 2} \right)\hspace{0.25in}\,\,y = \frac{1}{{t + 7}}\hspace{0.25in} - 1 \le t \le 2\)