Question

Using divergence theorem, evaluate the integral \(\iint_{S} \vec{F} \cdot \vec{n}\, dA\), where \(S\) is the surface of the cone \(x^{2}+y^{2} \le z^{2},\ 0 \le z \le 3\). If \(\vec{F} = 4x\hat{i} + 3z\hat{j} + 5y\hat{k}\) is a vector function with outer unit normal vector \(\vec{n}\), the value of the integral is ______ (rounded off to the nearest integer).

Hint:

For surfaces enclosing a volume, always try divergence theorem—usually far simpler than surface parametrization.

Answer 1

Correct Answer: 111

By the divergence theorem: \[ \iint_{S} \vec{F}\cdot\vec{n}\, dA = \iiint_{V} (\nabla\cdot \vec{F})\, dV \] Compute divergence: \[ \nabla\cdot \vec{F} = \frac{\partial}{\partial x}(4x) + \frac{\partial}{\partial y}(5y) + \frac{\partial}{\partial z}(3z) = 4 + 5 + 3 = 12 \] Thus the integral becomes: \[ \iiint_{V} 12\, dV = 12 \times \text{Volume of the cone} \] Volume of cone: \[ V = \frac{1}{3}\pi r^{2}h = \frac{1}{3}\pi (3^{2})(3) = 9\pi \] Flux value: \[ = 12 \times 9\pi = 108\pi \approx 339.29 \] Nearest integer: \[ \boxed{339\ \text{to}\ 341} \] Final Answer: 339–341