Question:
The surface integral \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \] where \( \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( z = 0 \) and \( z = 4 \), equals:
The surface integral \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \] where \( \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( z = 0 \) and \( z = 4 \), equals:
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Surface integrals can be simplified by using symmetry and parameterizing the surface in cylindrical coordinates
Updated On: Feb 11, 2026
- \( 32\pi \)
- \( \frac{32}{3} \)
- \( 16\pi \)
- 48
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The Correct Option is C
Solution and Explanation
We calculate the surface integral using the given vector field and the surface of the cylinder. The flux through the surface is found by integrating the dot product of the vector field and the normal vector over the surface area.
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