Problem $1$: Understanding of Discretization of Variables and Iterations

Consider the summation of the square of a continuous function $f(x)$ :

$y={\displaystyle \underset{i=1}{\overset{k}{}}}|f(x){|}^2$

where $x$ is a vector of length $k$ defined as a set of evenly$-$spaced values from $a$ to $b,$ i$.$e$.,$ in MATLAB we define: $x=l$ inspace $(a,b,k)$ and $k=1,2,3,$dots, etc.

a$)$ Develop a numerical solver $($i$.$e$.,$ function$)$ to execute the above algorithm and determine the maximum number of terms in the summation before it exceeds a user$-$specified value ymax. The solver should have the following input$/$output structure.

function $[y,N]=$ SquaredFunctionSum$($fun$,a,b,$ymax $)$

where, the output $N$ is the maximum number of terms in the series such that the sum does not exceed ymax.

b$)$ Test your developed numerical solver on the function $f(x)=x^3$ with inputs $a=1,b=3,$ ymax$=1000.$