9.5-bo'lim : Parametrik tenglamalar bilan sirt maydoni

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

1 – 4 masalalar uchun parametrik egri chiziqni berilgan o‘q atrofida aylantirish orqali olingan ob’ektning sirt maydonini aniqlang. Ushbu muammolar uchun siz \(t\) ning berilgan diapazoni uchun egri chiziq aynan bir marta chizilgan deb taxmin qilishingiz mumkin.

1. \(x = {t^2} - 3\hspace{0.25in}y = 2 + {t^2}\hspace{0.25in}0 \le t \le 5\)-ni \(x\) o'qi atrofida aylantiring.
2. \(x = - 8t\hspace{0.25in}y = 6 + {t^2}\hspace{0.25in} - 3 \le t \le 0\)-ni \(y\) o'qi atrofida aylantiring.
3. \(x = {t^2}\hspace{0.25in}y = {t^4} - 2\hspace{0.25in}0 \le t \le 2\)-ni \(y\) o'qi atrofida aylantiring.
4. \(x = 2 + t\hspace{0.25in}y = 4{{\bf{e}}^{ - \,\,\frac{1}{2}t}}\hspace{0.25in} - 1 \le t \le 2\)-ni \(x\) o'qi atrofida aylantiring.

5 - 7 muammolar uchun parametrik egri chiziqni berilgan o'q atrofida aylantirish natijasida olingan ob'ektning sirt maydonini beradigan integralni o'rnating, lekin baholamang. Ushbu muammolar uchun siz \(t\) ning berilgan diapazoni uchun egri chiziq aynan bir marta chizilgan deb taxmin qilishingiz mumkin.

1. \(x = 2 + {{\bf{e}}^{\cos \left( t \right)}}\hspace{0.25in}y = 1 + {t^2}\hspace{0.25in} - 2 \le t \le 0\)-ni \(x\) o'qi atrofida aylantiring.
2. \(x = {\cos ^2}\left( t \right)\hspace{0.25in}y = 2\cos \left( {2t} \right) - \sin \left( t \right)\hspace{0.25in}0 \le t \le 1\)-ni \(y\) o'qi atrofida aylantiring.
3. \(x = {t^2}\hspace{0.25in}y = \ln \left( {3 + {{\bf{e}}^{ - t}}} \right)\hspace{0.25in}0 \le t \le 2\)-ni \(x\) o'qi atrofida aylantiring.