Answer:
117
Step-by-step explanation:
45+39+33
<span>two and fifty-four hundredths </span>
You can parameterize
using spherical coordinates by
![\vec s(u,v)=\langle6\cos u\sin v,6\sin u\sin v,6\cos v\rangle](https://tex.z-dn.net/?f=%5Cvec%20s%28u%2Cv%29%3D%5Clangle6%5Ccos%20u%5Csin%20v%2C6%5Csin%20u%5Csin%20v%2C6%5Ccos%20v%5Crangle)
with
and
.
Take the normal vector to
to be
![\dfrac{\partial\vec s}{\partial\vec v}\times\dfrac{\partial\vec s}{\partial\vec u}=36\langle\cos u\sin^2v,\sin u\sin^2v,\cos v\sin v\rangle](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%5Cvec%20v%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%5Cvec%20u%7D%3D36%5Clangle%5Ccos%20u%5Csin%5E2v%2C%5Csin%20u%5Csin%5E2v%2C%5Ccos%20v%5Csin%20v%5Crangle)
(I use
to avoid negative signs. The orientation of the normal vector doesn't matter for a scalar surface integral; you could just as easily use
.)
Then
![f(x,y,z)=f(6\cos u\sin v,6\sin u\sin v,6\cos v)=36\sin^2v](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Df%286%5Ccos%20u%5Csin%20v%2C6%5Csin%20u%5Csin%20v%2C6%5Ccos%20v%29%3D36%5Csin%5E2v)
and the integral of
over
is
![\displaystyle\iint_Sf(x,y,z)\,\mathrm dS=\int_0^{\pi/2}\int_0^{2\pi}36\sin^2v\left\|\frac{\partial\vec s}{\partial v}\times\frac{\partial\vec s}{\partial u}\right\|\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_Sf%28x%2Cy%2Cz%29%5C%2C%5Cmathrm%20dS%3D%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B2%5Cpi%7D36%5Csin%5E2v%5Cleft%5C%7C%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Cright%5C%7C%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle\int_0^{\pi/2}\int_0^{2\pi}(36\sin^2v)(36\sin v)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint_0%5E%7B%5Cpi%2F2%7D%5Cint_0%5E%7B2%5Cpi%7D%2836%5Csin%5E2v%29%2836%5Csin%20v%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
If you get 4.4 interest in 1 year and 6 months is half of year, take half of 4.4 and multiply if by 550. Getting 1,210.