Question

T-Mobile 5:33 PM < Back MATH 1620 Pr… aビ Phase 2 In this phase, we will evaluate the integral numerically using the definition by Riemann sum. For numerical calculations, we will use MATLAB software. 3. First, use MATLAB to evaluate this time a definite integral. x ex dx = For that, type directly into command window in MATLAB: syms x; intx exp(x),0,2). Get the answer in a number with at least four decimals. Download an m-file, midPointRule.m, from Canvas and run it on MATLAB. Notice that the approximation is correct only up to two decimals (compare with the previous calculation). Increase the number of intervals up to 1000 and run the code one more time. Paste the figure you obtained as an answer for this question in your report and write the captions explaining what the figure exactly means Now you must modify the code to calculate the integral: 4. 5. You can use the same Midpoint Rule as in my code or, for extra points, modify it to use Trapezoidal Rule. Paste the figure and your code as an answer to the question. Dashboard Calendar To Do Notifications Inbox

Answer 1

Matlab code for integration clear all close all exact integration syms x I_ext-int (x*exp (x),0,2); function for which integration have to do f-e (x) x.*exp(x)i upper and lower limit a-0;b-2; displaying the function fprint f ( "\nFor the function having limit a-% f to b鵬f, f(x)-,,a,b) disp (f) fprintf( "\tExact Integration value for the function is %.\n',1-ext); %Integration using midpoint rule to get approximation correct upto two %dec imal s for n= 1 : 1000 I_mid-Midpoint_int (f,a,b,n) %checking wheather it correct upto two decimal or if abs ( 1-mid-1, ext )<=10^-3 not fprintf Integration value for the function using Midpoint is rule %f"\n',1 break mid); - end end syms x I_ext-int (x 2exp (x), 0,3) function for which integration have to do f-ê (x) x. 2. *exp(x); upper and lower limit a-0;b-3; fprint f ("AnFor the function having disp(f) fprint f ("AtExact integration value limit a-% f b.% f, f(x)-',a, b) to for the function is %.\n',1-ext); %Integration using Trapi zoidal rule to get approximation correct upto two decimals for n-1: 1000

I_trap-Trapizoidal_int (f,a,b,n); checking wheather it correct upto two decimal or not if abs(I_trap-I_ext)<-10A-3 fprintf Integration value for the function using Trapizoidal is rule %f.\n', l-trap ); break end end 888888888888888888888888888888888888888888钐8888888888888888888888 88 %%Matlab function for midpoint Method function val-Midpoint_int(f,a, b,n) 8 f is the function for integration % a is the lower limit of integration % b is the upper limit of integration % n is the number of trapizoidal interval in [a,b] dx-b-a)/n; %x interval val-0; 8splits interval a to b into n+1 subintervals x=linspace ( a,b,n+1) ; %loop for trapizoidal integration for i-1: n x mid(i)=(x(1+1)+x (i))/2; val-val+double(f (x mid(i)))i; end result using midpoint integration method val-dx*val; end Matlab function for Trap 12 01dal Method function val-Trapizoidal_int (f,a,b, n) f is the function for integration % a is the lower limit of integration % b is the upper limit of integration % n is the number of trapi zoidal interval in [a,b] dx= ( b-a )/n; %x interval val-0; splits interval a to b into n+1 subintervals xx-linspace (a,b, n+l); loop for trapizoidal integration for 1-2 : n val=va 1+ 2 * double (f(xx(i) )); end %Final integration value val for limit a to b val-(dx/2*(val+f (a)+f (b)) end 888888888888888888 End of Code 융8888888888888888

For the function having 1imit a-0.000000 to b-2.000000, f(x)- « (x)x. *exp(x) Exact Integration value for the function is 8.389056. Integration value for the function using Midpoint rule is 8.388076. the e(x)x.2.*exp(x) Exact Integration value for the function is 98.427685. 98.428682 function having limit a=0.000000 b=3.000000, f(x)- For to Integration value for the function using Trapizoidal rule is Published with MATLAB® R2018a

%%Matlab code for integration
clear all
close all

%exact integration
syms x
I_ext=int(x*exp(x),0,2);

%function for which integration have to do
f=@(x) x.*exp(x);
%upper and lower limit
a=0;b=2;
%displaying the function
fprintf('\nFor the function having limit a=%f to b=%f, f(x)=',a,b)
disp(f)
fprintf('\tExact Integration value for the function is %f.\n',I_ext);

%Integration using midpoint rule to get approximation correct upto two
%decimals

for n=1:1000
  
    I_mid=Midpoint_int(f,a,b,n);
    %checking wheather it correct upto two decimal or not
    if abs(I_mid-I_ext)<=10^-3
      
        fprintf('Integration value for the function using Midpoint rule is %f.\n',I_mid);
        break
      
    end
end

syms x
I_ext=int(x^2*exp(x),0,3);
%function for which integration have to do
f=@(x) x.^2.*exp(x);
%upper and lower limit
a=0;b=3;

fprintf('\nFor the function having limit a=%f to b=%f, f(x)=',a,b)
disp(f)
fprintf('\tExact Integration value for the function is %f.\n',I_ext);

%Integration using Trapizoidal rule to get approximation correct upto two
%decimals

for n=1:1000
  
    I_trap=Trapizoidal_int(f,a,b,n);
    %checking wheather it correct upto two decimal or not
    if abs(I_trap-I_ext)<=10^-3
      
        fprintf('Integration value for the function using Trapizoidal rule is %f.\n',I_trap);
        break
      
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%Matlab function for midpoint Method
function val=Midpoint_int(f,a,b,n)
    % f is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % n is the number of trapizoidal interval in [a,b]
    dx=(b-a)/n; %x interval
    val=0;
    %splits interval a to b into n+1 subintervals
    x=linspace(a,b,n+1);
    %loop for trapizoidal integration
        for i=1:n
            x_mid(i)=(x(i+1)+x(i))/2;
            val=val+double(f(x_mid(i)));
        end
    %result using midpoint integration method
    val=dx*val;
end

%%Matlab function for Trapizoidal Method
function val=Trapizoidal_int(f,a,b,n)
    % f is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % n is the number of trapizoidal interval in [a,b]
    dx=(b-a)/n; %x interval
    val=0;
    %splits interval a to b into n+1 subintervals
    xx=linspace(a,b,n+1);
    %loop for trapizoidal integration
        for i=2:n
            val=val+2*double(f(xx(i)));
        end
        %Final integration value val for limit a to b
        val=(dx/2)*(val+f(a)+f(b));
end
  
%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%