Biz chiziqli segment parametrlarini qanday olishni bilamiz, shuning uchun to'g'ridan-to'g'ri chiziq segmentining parametrlariga o'tamiz.

\[\vec r\left( t \right) = \left( {1 - t} \right)\left\langle {1,1,0} \right\rangle + t\left\langle {2,3, - 2} \right\rangle = \left\langle {1 + t,1 + 2t, - 2t} \right\rangle \,\,\,\,\,\,\,\,0 \le t \le 1\]

Biz buni quyidagi kabi parametr shakliga ham ajratishimiz mumkin.

\[\begin{array}{*{20}{l}}{x = 1 + t}\\{y = 1 + 2t}\\{z = - 2t}\end{array}\hspace{0.25in}0 \le t \le 1\]

Parametrlashtirishning har qanday shakli muammo uchun ishlaydi, ammo biz ushbu muammoning qolgan qismi uchun vektor shaklidan foydalanamiz.

Bizga parametrlash hosilasining kattaligi kerak bo'ladi, shuning uchun uni olamiz.

\[\vec r'\left( t \right) = \left\langle {1,2, - 2} \right\rangle \hspace{0.25in}\hspace{0.25in}\left\| {\vec r'\left( t \right)} \right\| = \sqrt {{{\left( 1 \right)}^2} + {{\left( 2 \right)}^2} + {{\left( { - 2} \right)}^2}} = \sqrt 9 = 3\]

Bizga parametrlashda "baholangan" integral ham kerak bo'ladi. Bularning barchasini eslang, biz integranddagi \(x\)/\(y\)ni parametrlashdan \(x\)/\(y\) bilan almashtiramiz. Bu erda parametrlashda baholangan integral.

\[xy - 4z = \left( {1 + t} \right)\left( {1 + 2t} \right) - 4\left( { - 2t} \right) = 2{t^2} + 11t + 1\]

U holda chiziqli integrali,

\[\int\limits_{C}{{xy - 4z\,ds}} = \int_{0}^{1}{{\left( {2{t^2} + 11t + 1} \right)\left( 3 \right)\,dt}} = \left. {3\left( {\frac{2}{3}{t^3} + \frac{{11}}{2}{t^2} + t} \right)} \right|_0^1 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{43}}{2}}}\]