8.1-bo'lim : Ark uzunligi

1. \(y = 14 - 9x\) , \( - 22 \le y \le 31\) uzunligi uchun integralni sozlang, lekin baholamang,
   1. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\)
   2. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy\)
2. \(x = {{\bf{e}}^{2y}}\), \( - 1 \le y \le 0\) uzunligi uchun integralni sozlang, lekin baholamang,
   1. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\)
   2. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy\)
3. \(y = \tan \left( {2x} \right)\), \(0 \le x \le \frac{\pi }{3}\) uzunligi uchun integralni sozlang, lekin baholamang,
   1. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\)
   2. \(\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy\)
4. \(\displaystyle \frac{{{x^2}}}{{16}} + 9{y^2} = 1\) uzunligi uchun integralni sozlang, lekin baholamang.
5. \(x = 6y + 1\) , \( - 2 \le y \le 8\) uchun
   1. Egri chiziq uzunligini topish uchun integraldan foydalaning.
   2. Javobingizni (a) qismdan geometrik tarzda tasdiqlang.
6. \(y = \frac{4}{3}x + 2\) , \(0 \le x \le 9\) uzunligini aniqlang.
7. \(y = {\left( {8x + 3} \right)^{\frac{3}{2}}}\) , \({11^{\frac{3}{2}}} \le y \le {27^{\frac{3}{2}}}\) uzunligini aniqlang.
8. \(x = {\left( {10 - 2y} \right)^{\frac{3}{2}}}\) , \( - 1 \le y \le 2\) uzunligini aniqlang.
9. \(x = 2 + {\left( {y - 1} \right)^2}\) , \(2 \le y \le 5\) uzunligini aniqlang.
10. \(y = {\left( {3x + 2} \right)^2}\) , \(1 \le x \le 4\) uzunligini aniqlang.