5.7-bo'lim : Aniq integrallarni hisoblash
17. Iloji bo'lsa, quyidagi integralni baholang. Agar buning iloji bo'lmasa, nima uchun integralni baholash mumkin emasligini aniq tushuntiring.
\[\int_{3}^{6}{{\left| {2x - 10} \right|\,dx}}\]Yechim
Ushbu integralni bajarish uchun biz mutlaq qiymat satrlarini "olib tashlashimiz" kerak. Biroq, buni amalga oshirish uchun biz \(2x - 10\) qayerda ijobiy va salbiy ekanligini bilishimiz kerak.
\(2x - 10\) chiziq tenglamasi bo'lgani uchun biz funktsiyaning quyidagi ijobiy/salbiy tabiatiga ega ekanligimiz aniq bo'lishi kerak.
\[\begin{align*}x & < 5\hspace{0.5in} \Rightarrow \hspace{0.5in}2x - 10 < 0\\ x & > 5\hspace{0.5in} \Rightarrow \hspace{0.5in}2x - 10 > 0\end{align*}\]2-qadam
Shunday qilib, mutlaq qiymat satrlarini olib tashlash uchun biz faqat \(x = 5\) da integralni buzishimiz kerak.
\[\int_{3}^{6}{{\left| {2x - 10} \right|\,dx}} = \int_{3}^{5}{{\left| {2x - 10} \right|\,dx}} + \int_{5}^{6}{{\left| {2x - 10} \right|\,dx}}\]Shunday qilib, birinchi integralda bizda \(3 \le x \le 5\) bor va shuning uchun birinchi integralda \(\left| {2x - 10} \right| = - \left( {2x - 10} \right)\) mavjud. Xuddi shunday, ikkinchi integralda bizda \(5 \le x \le 6\) bor va shuning uchun ikkinchi integralda \(\left| {2x - 10} \right| = 2x - 10\) mavjud. Yoki,
\[\int_{3}^{6}{{\left| {2x - 10} \right|\,dx}} = \int_{3}^{5}{{ - \left( {2x - 10} \right)\,dx}} + \int_{5}^{6}{{2x - 10\,dx}}\]3-qadam
Bu nuqtada qilishimiz kerak bo'lgan narsa - har bir integralni baholash. Mana o'sha ish.
\[\begin{align*}\int_{3}^{6}{{\left| {2x - 10} \right|\,dx}} & = \int_{3}^{5}{{ - 2x + 10\,dx}} + \int_{5}^{6}{{2x - 10\,dx}} = \left. {\left( { - {x^2} + 10x} \right)} \right|_3^5 + \left. {\left( {{x^2} - 10x} \right)} \right|_5^6\\ & = \left[ {25 - 21} \right] + \left[ { - 24 - \left( { - 25} \right)} \right] = \require{bbox} \bbox[2pt,border:1px solid black]{5}\end{align*}\]