GATE AG 2014 • Q37 MCQ • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Physical interpretations – line, surface and volume integrals

Question Statement:

The value of line integral $I = \int_C [(x^2y)\,dx + (x-z)\,dy + (xyz)\,dz]$, where $C$ is the arc of the parabola $y = x^2$ in the plane $z = 2$ from $(0, 0, 2)$ to $(1, 1, 2)$, is:

Options:

  • (A) −43/15
  • (B) 43/15
  • (C) −17/15
  • (D) 17/15

Official Answer Key:

C

Step-by-Step Solution & Explanation

Parametrize the arc: x = t, y = t², z = 2 (constant, so dz = 0), t: 0 → 1; dx = dt, dy = 2t dt. x²y dx = (t²)(t²) dt = t⁴ dt (x−z) dy = (t−2)(2t) dt = (2t² − 4t) dt xyz dz = 0 (since dz = 0) I = ∫₀¹ [t⁴ + 2t² − 4t] dt = [t⁵/5 + 2t³/3 − 2t²]₀¹ = 1/5 + 2/3 − 2 = 3/15 + 10/15 − 30/15 = −17/15. Answer: (C).

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