The GATE Grind

GATE 2025 IN – Question 36

Engineering Mathematics · Calculus and vector calculus · 2 marks · Multiple choice

The outward surface integral $$\iint_S(2x+z)\,dy\,dz+(2x+z)\,dx\,dz+(2z+y)\,dx\,dy$$ over the sphere $S:x^2+y^2+z^2=9$ is

  1. $72\pi$
  2. $144\pi$
  3. $36\pi$
  4. $432\pi$

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Correct answer: (B) $144\pi$

Explanation

Use the divergence theorem with $\mathbf F=(2x+z,2x+z,2z+y)$.

Its divergence is $2+0+2=4$.

The sphere has radius 3 and volume $(4/3)\pi(3)^3=36\pi$.

Thus the flux is $4(36\pi)=\boxed{144\pi}$, **B**.