8.2-bo'lim: Er yuzasi

1. \(x = \sqrt {y + 5} \) , \(\sqrt 5 \le x \le 3\) ni \(y\) o'qi atrofida aylantirish orqali olingan ob'ektning sirt maydoni uchun integralni o'rnating, lekin baholamang,
   1. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\)
   2. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy\)
   Yechim
2. \(y = \sin \left( {2x} \right)\) , \(\displaystyle 0 \le x \le \frac{\pi }{8}\) ni \(x\) o'qi atrofida aylantirish orqali olingan ob'ektning sirt maydoni uchun integralni o'rnating, lekin baholamang,
   1. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\)
   2. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy\)
   Yechim
3. \(y = {x^3} + 4\) , \(1 \le x \le 5\) ni berilgan o'q atrofida aylantirish orqali olingan ob'ektning sirt maydoni uchun integralni o'rnating, lekin baholamang. Siz ikkala ds dan foydalanishingiz mumkin.
   1. \(x\) o'qi
   2. \(y\) o'qi
   Yechim
4. \(y = 4 + 3{x^2}\) , \(1 \le x \le 2\) ni \(y\) o'qi atrofida aylantirish orqali olingan ob'ektning sirt maydonini toping. Yechim
5. \(y = \sin \left( {2x} \right)\) , \(\displaystyle 0 \le x \le \frac{\pi }{8}\) ni \(x\) o'qi atrofida aylantirish orqali olingan ob'ektning sirt maydonini toping. Yechim