GATE 2022 BM – Question 51
Using the divergence theorem, evaluate outward flux $\iint_S\mathbf F\cdot\mathbf n\,dA$ across the closed surface of $x^2+y^2\le z^2,0\le z\le3$, for $\mathbf F=4x\hat i+3z\hat j+5y\hat k$. Round to the nearest integer.
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Correct answer: 111 to 114
Explanation
The divergence is ∂(4x)/∂x+∂(3z)/∂y+∂(5y)/∂z=4. The cone has radius three and height three, hence volume π×3²×3/3=9π. The closed outward flux is divergence times volume=36π≈113.097, rounded to 113.