Question

1)

a) Write MATLAB function that accepts a positive integer parameter n and returns a vector containing the values of the integral (A) for n= 1,2,3,..., n. The function must use the relation (B) and the value of y(1). Your function must preallocate the array that it returns. Use for loop when writing your code.

b) Write MATLAB script that uses your function to calculate the values of the integral (A) using the recurrence relation (B), y(n) for n=1,2,... 19

2. The goal of this exercise is (a) al of this exercise is (a) to write a matlab function that uses for loop and preallo- as an array, and (b) to write a matlab script that plots graphs of functions. Consider the following integral: y(n) = f *"e-*dx, (A) where n is non-negative integer, n = 0,1,2,... The integral (A) has the following properties: 1. All y(n) are positive, y(n) > 0 for n = 0,1,.. 2. y(n) is monotonously decreasing function of n, i.e. y(n) < y(n-1) for n = 1,2..... To evaluate the integral numerically, we integrate (A) by parts to obtain the followi. relation: y(n) = nyin- The value of the integral (A) for n=0 is (B) y(0) = ſexdx=1 - (C) Using (C) and (B) for n = 1 we can find that y(1) = y(0) - = 1 -(D We now can repeat the steps to calculate y(2), y(3),... Page 2 of 4

Answer 1

%&Matlab code integral clear all close all loop for finding integration using A fprintf('Integration using formula A\n') for i=1:19 val=int_fun(i); fprintf('\tFor n=$d the integral value is $\n',i,val) end #loop for finding integration using B in_val(1)=1-1/exp(1); fprintf('Integration using formula B\n') for i=1:19 val=i* in_val(i)-(1/exp(1)); in_val(i+1)=val; fprintf('\tFor n=%d the integral value is f\n',i,val) end function for integration function val=int_fun(n) fun=@(x) x. n. *exp(-x); val=integral(fun,0,1); end %%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%% Integration using formula A For n=1 the integral value is 0.264241 For n=2 the integral value is 0.160603 For n=3 the integral value is 0.113929 For n=4 the integral value is 0.087836 For n=5 the integral value is 0.071302 For n=6 the integral value is 0.059934 For n-7 the integral value is 0.051656 For n=8 the integral value is 0.045368 For n-9 the integral value is 0.040434 For n=10 the integral value is 0.036461 For n=11 the integral value is 0.033195 For n=12 the integral value is 0.030463 For n=13 the integral value is 0.028145 For n=14 the integral value is 0.026154 For n=15 the integral value is 0.024424 For n=16 the integral value is 0.022909 For n=17 the integral value is 0.0215 70 For n=18 the integral value is 0.020378 For n=19 the integral value is 0.019311 Integration using formula B

For n=1 the integral value is 0.264241 For n=2 the integral value is 0.160603 For n=3 the integral value is 0.113929 For n=4 the integral value is 0.087836 For n=5 the integral value is 0.071302 For n=6 the integral value is 0.059934 For n=7 the integral value is 0.051656 For n=8 the integral value is 0.045368 For n=9 the integral value is 0.040434 For n-10 the integral value is 0.036461 For n=11 the integral value is 0.033195 For n-12 the integral value is 0.030463 For n=13 the integral value is 0.028146 For n=14 the integral value is 0.026159 For n=15 the integral value is 0.024505 For n=16 the integral value is 0.024198 For n=17 the integral value is 0.0434 80 For n=18 the integral value is 0.414758 For n=19 the integral value is 7.512522 Published with MATLAB® R2018a

%%Matlab code integral
clear all
close all

%loop for finding integration using A
fprintf('Integration using formula A\n')
for i=1:19
    val=int_fun(i);
    fprintf('\tFor n=%d the integral value is %f\n',i,val)
end

%loop for finding integration using B
in_val(1)=1-1/exp(1);

fprintf('Integration using formula B\n')
for i=1:19
    val=i*in_val(i)-(1/exp(1));
    in_val(i+1)=val;
    fprintf('\tFor n=%d the integral value is %f\n',i,val)
end

%function for integration
function val=int_fun(n)

        fun=@(x) x.^n.*exp(-x);
        val=integral(fun,0,1);
      
end

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%