Asosiy tarkibga o'tish

7.10-bo'lim : Aniq integrallarni taxmin qilish

3. \(n = 8\) dan foydalanib, \(\displaystyle \int_{0}^{4}{{\cos \left( {1 + \sqrt x } \right)\,dx}}\) qiymatini taxminiy hisoblang

  1. O'rta nuqta qoidasi,
  2. Trapezoid qoidasi va
  3. Simpson qoidasi.

Ishingiz uchun kamida 6 kasrdan iborat aniqlikdan foydalaning.

a O'rta nuqta qoidasi Yechim

Muammoni hal qilish uchun haqiqatan ham kerak bo'lmasa-da, bu erda grafikning eskizi.

The graph of \(\cos \left( {1 + \sqrt x } \right)\) on the interval \(\left[ {0,4} \right]\). The graph starts at about 0.54, drops steeply and crosses the x-axis just after \(x = 0\), then continues decreasing more and more slowly down to about -1 at the right end. The nine points that divide the interval into 8 subintervals are marked along the x-axis.

Biz \(\left[ {0,4} \right]\) oralig'ini har birining kengligi bo'lgan 8 ta kichik intervalgacha bo'lishimiz kerakligini bilamiz,

\[\Delta x = \frac{{4 - 0}}{8} = \frac{1}{2}\]

Ushbu kichik intervallarning har birining so'nggi nuqtalari yuqoridagi grafikdagi \(x\) o'qidagi nuqtalar bilan ifodalanadi.

Har bir nuqta orasidagi belgi har bir subintervalning o'rta nuqtasini ifodalaydi. Har bir kichik oraliq uchun o'rta nuqtalarning \(x\) qiymatlari quyidagicha:

\[\frac{1}{4},\hspace{0.25in}\frac{3}{4},\hspace{0.25in}\frac{5}{4},\hspace{0.25in}\frac{7}{4},\hspace{0.25in}\frac{9}{4},\hspace{0.25in}\frac{{11}}{4},\hspace{0.25in}\frac{{13}}{4},\hspace{0.25in}\frac{{15}}{4}\]

Shunday qilib, integralning qiymatini taxmin qilish uchun O'rta nuqta qoidasidan foydalanish uchun formulaga ulanishimiz kerak. Shunday qilib,

\[\begin{align*}\int_{0}^{4}{{\cos \left( {1 + \sqrt x } \right)\,dx}} & \approx \left( {\frac{1}{2}} \right)\left[ {f\left( {\frac{1}{4}} \right) + f\left( {\frac{3}{4}} \right) + f\left( {\frac{5}{4}} \right) + f\left( {\frac{7}{4}} \right) + f\left( {\frac{9}{4}} \right) + } \right.f\left( {\frac{{11}}{4}} \right) + \\ & \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\left. {f\left( {\frac{{13}}{4}} \right) + f\left( {\frac{{15}}{4}} \right)} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 2.51625938}}\end{align*}\]

b Trapetsiya qoidasi Yechim

Midpoint Rule ishidan biz bilamizki, har bir kichik oraliqning kengligi \(\Delta x = \frac{1}{2}\) va mos yozuvlar uchun grafikning eskizi va nuqtalar bilan belgilangan har bir subintervalning so'nggi nuqtalari quyida ko'rsatilgan.

The graph of \(\cos \left( {1 + \sqrt x } \right)\) on the interval \(\left[ {0,4} \right]\). The graph starts at about 0.54, drops steeply and crosses the x-axis just after \(x = 0\), then continues decreasing more and more slowly down to about -1 at the right end. The nine points that divide the interval into 8 subintervals are marked along the x-axis.

Shunday qilib, integralning qiymatini taxmin qilish uchun Trapezoid qoidasidan foydalanish uchun formulaga ulanishimiz kerak. Shunday qilib,

\[\begin{align*}\int_{0}^{4}{{\cos \left( {1 + \sqrt x } \right)\,dx}} & \approx \left( {\frac{{{\scriptstyle{}^{1}/{}_{2}}}}{2}} \right)\left[ {f\left( 0 \right) + 2f\left( {\frac{1}{2}} \right) + 2f\left( 1 \right) + 2f\left( {\frac{3}{2}} \right) + 2f\left( 2 \right) + 2} \right.f\left( {\frac{5}{2}} \right) + \\ & \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\left. {2f\left( 3 \right) + 2f\left( {\frac{7}{2}} \right) + f\left( 4 \right)} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 2.43000475}}\end{align*}\]

c Simpson qoidasi Yechim

Midpoint Rule ishidan biz bilamizki, har bir kichik oraliqning kengligi \(\Delta x = \frac{1}{2}\) va mos yozuvlar uchun grafikning eskizi va nuqtalar bilan belgilangan har bir subintervalning so'nggi nuqtalari quyida ko'rsatilgan.

The graph of \(\cos \left( {1 + \sqrt x } \right)\) on the interval \(\left[ {0,4} \right]\). The graph starts at about 0.54, drops steeply and crosses the x-axis just after \(x = 0\), then continues decreasing more and more slowly down to about -1 at the right end. The nine points that divide the interval into 8 subintervals are marked along the x-axis.

Birinchi ikki qismda bo'lgani kabi, integralning taxminiy qiymatini aniqlash uchun Simpson qoidasidan foydalanish uchun formulaga ulanishimiz kerak. Shunday qilib,

\[\begin{align*}\int_{0}^{4}{{\cos \left( {1 + \sqrt x } \right)\,dx}} & \approx \left( {\frac{{{\scriptstyle{}^{1}/{}_{2}}}}{3}} \right)\left[ {f\left( 0 \right) + 4f\left( {\frac{1}{2}} \right) + 2f\left( 1 \right) + 4f\left( {\frac{3}{2}} \right) + 2f\left( 2 \right) + 4} \right.f\left( {\frac{5}{2}} \right) + \\ & \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\left. {2f\left( 3 \right) + 4f\left( {\frac{7}{2}} \right) + f\left( 4 \right)} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 2.47160136}}\end{align*}\]