Answer:
Inequality: x > -2
Interval Notation: ( -2, ∞)
Step-by-step explanation:
Apparently my answer was unclear the first time?
The flux of <em>F</em> across <em>S</em> is given by the surface integral,
![\displaystyle\iint_S\mathbf F\cdot\mathrm d\mathbf S](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_S%5Cmathbf%20F%5Ccdot%5Cmathrm%20d%5Cmathbf%20S)
Parameterize <em>S</em> by the vector-valued function <em>r</em>(<em>u</em>, <em>v</em>) defined by
![\mathbf r(u,v)=7\cos u\sin v\,\mathbf i+7\sin u\sin v\,\mathbf j+7\cos v\,\mathbf k](https://tex.z-dn.net/?f=%5Cmathbf%20r%28u%2Cv%29%3D7%5Ccos%20u%5Csin%20v%5C%2C%5Cmathbf%20i%2B7%5Csin%20u%5Csin%20v%5C%2C%5Cmathbf%20j%2B7%5Ccos%20v%5C%2C%5Cmathbf%20k)
with 0 ≤ <em>u</em> ≤ π/2 and 0 ≤ <em>v</em> ≤ π/2. Then the surface element is
d<em>S</em> = <em>n</em> • d<em>S</em>
where <em>n</em> is the normal vector to the surface. Take it to be
![\mathbf n=\dfrac{\frac{\partial\mathbf r}{\partial v}\times\frac{\partial\mathbf r}{\partial u}}{\left\|\frac{\partial\mathbf r}{\partial v}\times\frac{\partial\mathbf r}{\partial u}\right\|}](https://tex.z-dn.net/?f=%5Cmathbf%20n%3D%5Cdfrac%7B%5Cfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20v%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20u%7D%7D%7B%5Cleft%5C%7C%5Cfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20v%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20u%7D%5Cright%5C%7C%7D)
The surface element reduces to
![\mathrm d\mathbf S=\mathbf n\,\mathrm dS=\mathbf n\left\|\dfrac{\partial\mathbf r}{\partial u}\times\dfrac{\partial\mathbf r}{\partial v}\right\|\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cmathbf%20S%3D%5Cmathbf%20n%5C%2C%5Cmathrm%20dS%3D%5Cmathbf%20n%5Cleft%5C%7C%5Cdfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cmathbf%20r%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![\implies\mathbf n\,\mathrm dS=-49(\cos u\sin^2v\,\mathbf i+\sin u\sin^2v\,\mathbf j+\cos v\sin v\,\mathbf k)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cimplies%5Cmathbf%20n%5C%2C%5Cmathrm%20dS%3D-49%28%5Ccos%20u%5Csin%5E2v%5C%2C%5Cmathbf%20i%2B%5Csin%20u%5Csin%5E2v%5C%2C%5Cmathbf%20j%2B%5Ccos%20v%5Csin%20v%5C%2C%5Cmathbf%20k%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
so that it points toward the origin at any point on <em>S</em>.
Then the integral with respect to <em>u</em> and <em>v</em> is
![\displaystyle\iint_S\mathbf F\cdot\mathrm d\mathbf S=\int_0^{\pi/2}\int_0^{\pi/2}\mathbf F(x(u,v),y(u,v),z(u,v))\cdot\mathbf n\,\mathrm dS](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_S%5Cmathbf%20F%5Ccdot%5Cmathrm%20d%5Cmathbf%20S%3D%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cmathbf%20F%28x%28u%2Cv%29%2Cy%28u%2Cv%29%2Cz%28u%2Cv%29%29%5Ccdot%5Cmathbf%20n%5C%2C%5Cmathrm%20dS)
![=\displaystyle-49\int_0^{\pi/2}\int_0^{\pi/2}(7\cos u\sin v\,\mathbf i-7\cos v\,\mathbf j+7\sin u\sin v\,\mathbf )\cdot\mathbf n\,\mathrm dS](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle-49%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B%5Cpi%2F2%7D%287%5Ccos%20u%5Csin%20v%5C%2C%5Cmathbf%20i-7%5Ccos%20v%5C%2C%5Cmathbf%20j%2B7%5Csin%20u%5Csin%20v%5C%2C%5Cmathbf%20%29%5Ccdot%5Cmathbf%20n%5C%2C%5Cmathrm%20dS)
![=-343\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\cos^2u\sin^3v\,\mathrm du\,\mathrm dv=\boxed{-\frac{343\pi}6}](https://tex.z-dn.net/?f=%3D-343%5Cdisplaystyle%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B%5Cpi%2F2%7D%5Ccos%5E2u%5Csin%5E3v%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cboxed%7B-%5Cfrac%7B343%5Cpi%7D6%7D)
The correct answer would be 1.