GATE 2017 EC – Question 38
Let $I = \int_C (2z\,dx + 2y\,dy + 2x\,dz)$ where $x, y, z$ are real, and let $C$ be the straight line segment from point $A: (0, 2, 1)$ to point $B: (4, 1, -1)$. The value of $I$ is ________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: -11
Explanation
Parametrise the line as $x = 4t$, $y = 2 - t$, $z = 1 - 2t$ for $0 \leq t \leq 1$. Then $dx = 4\,dt$, $dy = -dt$ and $dz = -2\,dt$. The integrand becomes $2(1 - 2t)(4) + 2(2 - t)(-1) + 2(4t)(-2) = 8 - 16t - 4 + 2t - 16t = 4 - 30t$. Integrating from 0 to 1 gives $4 - 15 = -11$.