a$)$ Write a program to implement the midpoint approximation in two dimensions and integrate the function $f(x,y)=3x^2+2xy+y^2$ over the region defined by the condition $x^2+y^{2}0\mathrm{.}4.$ Use $h=0\mathrm{.}1,0\mathrm{.}05,0\mathrm{.}025,$ and $0\mathrm{.}0125.$

b$)$ Repeat part a$)$ using a Monte Carlo method and the same number of points $n.$ For each value of $n,$ repeat the calculation several times to obtain a crude estimate of the random error.

c$)$ Discuss the differences in the errors of the results in part a$)$ and b$).$