Integrate the vector function
$$F=(2x)\hat{i}+(4y)\hat{j}-(5z)\hat{k}$$ Over the closed surfaces of the volume defined between the surfaces of
$x^2+y^2+z^2=4$ and $x^2+y^2+z^2=1$ and $z>0$.
I used Green's divergence method but I couldn't able to reach a meaningful solution.
Since you have a closed surface, the divergence theorem would be appropriate.
$\nabla\cdot F = 1$
$\iiint dV = V$
Which is the volume between two hemi-spheres.
$\frac 23 \pi (R^3 - r^3) = \frac {14\pi}{3}$
$\int_0^{2\pi}\int_{0}^{\frac {\pi}{2}}\int_1^2 \rho^2\sin\phi \;d\rho\;d\phi\;d\theta\\\int_0^{2\pi}\int_{0}^{\frac {\pi}{2}}\frac 13 \rho^3\sin\phi| _1^2\;d\phi\;d\theta\\ \int_0^{2\pi}\int_{0}^{\frac {\pi}{2}}\frac 13 (8-1)\sin\phi \;d\phi\;d\theta\\ \int_0^{2\pi}\int_{0}^{\frac {\pi}{2}}\frac 73 \sin\phi \;d\phi\;d\theta\\ \int_0^{2\pi}\frac 73 (-\cos\phi)|_0^{\frac\pi 2} \;d\theta\\ \int_0^{2\pi}\frac 73 \;d\theta\\ \frac 73 (2\pi)\\ \frac {14\pi}3$