# Application of the Gauss's Divergence Theorem

Question: Let $$S$$ be the closed surface forming the boundary of the region $$V$$ bounded by $$x^2+y^2=3$$, $$z=0,\ z=6$$. A vector field $$\vec{F}$$ is defined over $$V$$ with $$\nabla.\vec{F}=2y+z+1$$. What is the value of $$\displaystyle\frac{1}{\pi} \iint_S{\vec{F}. \hat{n}dS}$$ ?

Attempt: We calculate $$\displaystyle \frac{1}{\pi}\int_{-\sqrt{3}}^{+\sqrt{3}}\int_{-\sqrt{3-x^2}}^{+ \sqrt{3-x^2}}\int_0^6 {\nabla.\vec{F}dzdydx}$$ $$=\displaystyle \frac{1}{\pi}\int_{-\sqrt{3}}^{+\sqrt{3}}\int_{-\sqrt{3-x^2}}^{+ \sqrt{3-x^2}}12(2+y)dydx$$. Converting to polar:

$$\displaystyle \frac{1}{\pi}\int_{0}^{2\pi}\int_{0}^{\sqrt{3}}12(2+rsin(\theta))rdrd\theta=72$$.

Did I make any mistake? The answer provided is $$24.61$$.