Chapter 14.FOM, Problem 6P

Areas of Curved Regions The Monte Carlo method can be used to estimate the area under the graph of a function. The figure below shows the region under the graph of $f\left(x\right)=x^2$, above the $x$-axis, between $x=0$ and $x=1$. If we choose a point in the square at random, the probability that it lies under the graph of $f\left(x\right)=x^2$ is the area under the graph divided by the area of the square. So if we randomly select a large number of point in the square, we have

$\frac{\text{number of hits under the graph}}{\text{number of hits in the square}}\approx \frac{\text{area under graph}}{\text{area}\text{of}\text{square}}$

Modify the program from Problem $5$ to carry out this Monte Carlo simulation and approximate the required area.