DefnOfDefiniteIntegral - 8-masala yechimi | MathModel.uz

5.6-bo'lim : Aniq integralning ta'rifi

8. $ \displaystyle \int_{1}^{4}{{3x - 2\,dx}}$ uchun integralning grafigini chizing va integralning qiymatini aniqlash uchun aniq integralning maydon talqinidan foydalaning.

Yechim

Mana $\left[ {1,4} \right]$ intervalidagi $f\left( x \right) = 3x - 2$ integralining grafigi.

$The graph of the integrand $f\left( x \right) = 3x - 2$ on the interval $\left[ {1,4} \right]$. It is a straight line rising from $\left( {1,1} \right)$ to $\left( {4,10} \right)$ and the region between the line and the x-axis is shaded in.$

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. Mana bunga yordam beradigan "o'zgartirilgan" grafik.

$The graph of the integrand $f\left( x \right) = 3x - 2$ on $\left[ {1,4} \right]$ with the shaded area split up into two simpler pieces. A horizontal dashed line has been drawn from $\left( {1,1} \right)$ to $\left( {4,1} \right)$, which splits the shaded region into a rectangle of width 3 and height 1 along the bottom and a triangle of base 3 and height 9 sitting on top of it. The points $\left( {1,1} \right)$, $\left( {4,1} \right)$ and $\left( {4,10} \right)$ are marked and labeled.$

Ushbu eskizdan biz bu maydonni kengligi 3 va balandligi 1 bo'lgan to'rtburchaklar va asosi 3 va balandligi 9 bo'lgan uchburchak sifatida tasavvur qilishimiz mumkinligini ko'rishimiz mumkin. Keyin integralning qiymati to'rtburchak va uchburchak maydonlarining yig'indisiga teng bo'ladi.

Bu erda integralning qiymati,

\[\int_{1}^{4}{{3x - 2\,dx}} = \left( 3 \right)\left( 1 \right) + \frac{1}{2}\left( 3 \right)\left( 9 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{33}}{2}}}\]