IN MAT MATLAB USE TEMPLATE GIVEN

function project$\_12()\%$ PROJECT$\_12$ project$\_12$ is the driver function for the program.

Name: Date: Class:

$\%$

functior

$\%$

Implement Monte Carlo Method to Integrate

$\%$ Print the splash screen using the print$\_$header function

$\%$ Use the get$\_$function function to enter the integrand

$\%$ Call get$\_$data three times. The three outputs will be

$\%$ Call get$\_$data three times. The three outputs will be $\%$ a $-$ the lower bound for the integral $($must be greater than $\%$ b $-$ the upper bound for the integral $($must

$\%$ than $-$ number of elements in the vectors $($must be greater

end

function plot$\_$results$($f$,x,y,$ hits$)\%$ PLOTRESULTS plotesults hits$)$ is a function to $\%$ create a plot of the integral. It does this by plotting a $\%$ create a plot of the integral. It does this by plotting a green dot $\%($values in pHit or $n$ Hit$)$ for each point between the function and the $\%$ horizontal axis. It then plots a red dot at each point $($redvector$)$

$\%$ where the point

Objective Use simulation modeling to approximate the value of a definite integral.

Introduction In the Calculus we study integration and in particular definite integrals. With this comes many techniques for evaluating integrals. Sadly after studying substitutions, integration by parts, partial fractions, and infinite series techniques we discover that most functions simply cannot be integrated in a closed form. Instead we must opt for an approximation method to evaluate the definite integral. One such approximation technique is a Monte Carlo method. In Monte Carlo methods we use a computer to generate random numbers and then use those random numbers to approximate a mathematical model $-$ in this case a definite integral.

How does Monte Carlo integration work? The technique with which integration is presented is to calculate the area under the curve. This is usually done by fitting rectangles or other geometric shapes between the graph of the function and the $x-$axis. The areas of these shapes are then added together and the sum is an approximation of the definite integral. Monte Carlo integration works in almost the same way. The difference is that instead of shapes we will fill the area with points and count the number of points that are between the curve and the horizontal axis.

Methodology While the technique may sound a bit vague the implementation is straightforward and in MatLab it can be programmed without loops. The steps are

Have the user enter an anonymous function using the get$\_$function function.

Have the user enter the lower bound, $A,$ and the upper bound, $B,$ for the integration.

To start the Monte Carlo modeling generate a vector of random numbers for $x.$ These values must be uniformly distributed between A and $B.$ The MatLab command is

$x=(B-A)**rand(1,n)+A$;

In this command $n$ is the number of random $x$ values that you would like to generate.

$4.$ Now Have the program find the minimum value of the user entered function on the interval $A,B.$ Call this scalar $m.$ You can either either use the fminbnd or min functions to find the value. Since you already have a vector of input values using $)$ works well for this. There is an extra step here $-$ if $m$ is greater than zero, then $m$ must be set to zero.

$5.$ Repeat this but this time have the program find the maximum value of the function on the interval $A,B.$ Call this scalar $M.$ As before, you can either either use the fminbnd or max functions. To use fminbnd you pass the negative of the function, $-f(x),$ the the function, as such it is probably easier to use the max function using your already created $x$ vector. Similarly to the minimum, if $M$ is less than zero then it must be set to zero.

$6.$ These four points create a rectangle whose corners are $(A,m),(B,m),(B,M)$ and $(A,M).$ The area of this rectangle is

Area $=(B-A)(M-m)$