How do I calculate vector fields $\mathbf F(x,y,z)=x\mathbf i+y\mathbf j+ z\mathbf k$ flux through the cylinder $S =${$ (x,y,z)\vert x^2+y^2\le 9, 0\le z \le 2$}.
I know that:
$$dr=(-3\sin\theta,3\cos\theta,0)$$
$\Phi=\int \mathbf F \cdot d^2A=\int_0^{2\pi}\int_0^2(3\cos\theta,3\sin\theta,z)\cdot(3\cos\theta,3\sin\theta,0)\,dzd\theta=\int_0^{2\pi}\int_0^29\,dzd\theta=36\pi$
Right answer is $54\pi$. I don't know where I have done mistake but it's so close that tehre are some dump mistakes.
I'd use Gauss' (Divergence) Theorem here. The divergence of the vector field is $3$, which you integrate over the volume of the cylinder $18\pi.$ So you get $3(18\pi.)$ Much less painful.