# Surface integral over a discontinuous flux

Calculus Level 5

Find the flux $$A$$ over the surface region of the sphere of radius 5, centred in origin, limited by the plans $$z=3$$ and $$z=-4$$, where the field is: $$F(x,y,z)=\left(\frac{y}{\sqrt{I}},\dfrac{-x}{\sqrt{I}}+7y,-7z\right)$$

$${I}=(4-\sqrt{x^{2}+y^{2}})^{2}+z^{2}$$

For that, do you will need to find the flux over the torus of radius $$R=4$$ and $$r=\epsilon$$, centred in origin, and make the limit:

$$\displaystyle B= \lim_{\epsilon \to 0} \int \int_T F(x,y,z)\cdot \hat{n_{in}} \,dS.$$

Your answer for $$A$$ should be something like:

$$\displaystyle A=a\pi+\lim_{\epsilon \to 0} (b\pi^{2} \epsilon^{2} )$$

Submit $$a+b$$ as your answer.

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