# Gauss divergence theorem

I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux through a sphere. The sphere is given as $$x^2 +y^2+z^2==4$$ where as the z is resticted from $$\sqrt{3} \; to \; 2$$ I determined the divergence to $$4z$$ I first tried to use spherical coordinates which resolve to:

$$\int _{\sqrt{3}}^2\int _0^{\pi /6}\int _0^{2\pi }4 *r*\text{Cos}[o]*r^2*\text{Sin}[o]dpdodr = \frac{7 \pi }{4}$$

To confirm this solution I tried to solve this problem using a regular volume integral. Which resolve to: $$\int _{\sqrt{3}}^2\int _{-\sqrt{4-z^2}}^{\sqrt{4-z^2}}\int _{-\sqrt{4-y^2-z^2}}^{\sqrt{4-y^2-z^2}}(4 *z) dxdydz = \pi$$ I guess the error lies in my changed z variable. I replaced it with $$r*\text{Cos}[o]$$ Could you guys please take a look into it? Thanks

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