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:

• A. \( 32\pi \)
• B. \( \frac{32}{3} \)
• C. \( 16\pi \)
• D. 48

Hint:

Surface integrals can be simplified by using symmetry and parameterizing the surface in cylindrical coordinates

Answer 1

The Correct Option is C

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.