Solve in Matlab.

The methods that we used for approximating single integrals all have counterparts for double integrals. The "Midpoint Rule" for double integrals is given in the following equation.

${}_{R}f(x,y)dA$~~${\displaystyle \underset{i=1}{\overset{m}{}}}{\displaystyle \underset{j=1}{\overset{n}{}}}f({\stackrel{}{x}}_i\frac{,}{b}ar(y{)}_j)xy$

where $\frac{?}{\text{b}\text{a}\text{r}}(x_i)$ is the midpoint of $x_{i-1},x_i$ and $\frac{?}{\text{b}\text{a}\text{r}}(y_j)$ is the midpoint of $y_{j-1},y_j$

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For example, eq$.(1)$ can be applied to estimate the volume of the solid $($see Fig. $1)$ that lies above the square $R=[0,2]x[0,2]($see Fig. $2)$ and below the elliptic paraboloid $f(x,y)=16-x-2y^2.$

Write a local function that will return the final value of the double integral given in $(1).$ Start with the function definition shown below. The function will receive $6$ input arguments:

the original function $f(x,y),$

the lower bound of the double integral in the $x-$direction $(a),$

the upper bound of the double integral in the $x-$direction $(b),$

the lower bound of the double integral in the $y-$direction $($c$),$

the upper bound of the double integral in the $y-$direction $(d),$

number of sub$-$intervals $(n).$

We assume that the number of sub$-$intervals are the same in both $x$ and $y$ direction.