Use a user$-$defined Simpson's rule function to solve the problems below. Do this in a single script file and report the results for each part of the problem using fprintf. Make sure what you print out makes sense and includes appropriate units.

$($a$)$ The variation of gravitational acceleration, $g,$ with altitude, $y,$ is given by

$g=\frac{R^2}{(R+y{)}^2}{g}_0$

where $R=6371km$ is the radius of the Earth, and $g_0=9\mathrm{.}81\frac{m}{s^2}.$ The change in gravitational potential energy, $U,$ is given by

$U={\displaystyle \underset{0}{\overset{h}{}}}mgdy$

Determine the change in the potential energy of a satellite with a mass of $500kg$ that is raised from the surface of the Earth to a height of $800km.$

$($b$)$ The orbit of Pluto is elliptical, with a semi$-$major axis of $a=5\mathrm{.}9065{10}^{9}km$ and a semi$-$minor axis of $b=5\mathrm{.}7208{10}^{9}km.$ The perimeter of the ellipse can be calculated with the integral

$P=4a{\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}\sqrt[\phantom{2}]{1-k^{2}si{n}^2}d$

where $k=\frac{\sqrt[\phantom{2}]{a^2-b^2}}{a}.$ Determine the distance Pluto travels in one orbit and calculate the average speed at which Pluto travels $($in $k\frac{m}{h})$ if one orbit takes about $248$ year.

$($c$)$ Do the same integrals using MATLAB's built$-$in function integral. Compare the results.