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GATE 2014 EE (EE1) – Question 38

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 2 marks · Multiple choice

Find the counterclockwise line integral of $\mathbf F=yz\hat i$ around $x^2+y^2=1$ at z = 1.

  1. $-2\pi$
  2. $-\pi$
  3. $\pi$
  4. $2\pi$

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Correct answer: (B) $-\pi$

Explanation

At z=1parameterize the unit circle by x=cosθ,y=sinθfor0≤θ≤2π. Then F=(sinθ,0,0)and dl=(−sinθ,cosθ,0)dθ. Their dot product is−sin²θdθ, whose integral is **−π,B**. Counterclockwise is viewed from positive z.