5.6-bo'lim : Aniq integralning ta'rifi
9. \( \displaystyle \int_{0}^{5}{{ - 4x\,dx}}\) uchun integralning grafigini chizing va integralning qiymatini aniqlash uchun aniq integralning maydon talqinidan foydalaning.
Yechim
Mana \(\left[ {0,5} \right]\) intervalidagi \(f\left( x \right) = - 4x\) integralining grafigi.
![The graph of the integrand \(f\left( x \right) = - 4x\) on the interval \(\left[ {0,5} \right]\). It is a straight line falling from the origin down to \(\left( {5, - 20} \right)\), which is marked and labeled, and the region between the line and the x-axis is shaded in. The whole region lies below the x-axis.](/Solutions/CalcI/DefnOfDefiniteIntegral//Solutions/CalcI/DefnOfDefiniteIntegral/DefnofDefiniteIntegral_Files/image003.webp)
2-qadam
Endi biz bilamizki, integral shunchaki chiziq va \(x\) o'qi orasidagi maydondir va shuning uchun biz integralning qiymatini aniqlashga yordam berish uchun asosiy maydon formulalaridan foydalanishimiz kerak.
Bu holda biz maydonning asosi 5 va balandligi 20 bo'lgan uchburchak ekanligini aniq ko'rishimiz mumkin. Biroq, biz bu erda biroz ehtiyot bo'lishimiz va \(x\) o'qidan pastda joylashgan maydon salbiy maydon deb hisoblanishini esga olishimiz kerak, shuning uchun biz maydonni hisoblashda buni yodda tutishimiz kerak.
Bu erda integralning qiymati,
\[\int_{0}^{5}{{ - 4x\,dx}} = - \frac{1}{2}\left( 5 \right)\left( {20} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 50}}\]