Question

1. Approximate the following integral, exp(r) using the composite midpoint rule, composite trapezoid rule, and composite Simpeon's method. Each method should invol + l integrand evaluations, k 1: 20. On the same plot, graph the absolute error as a function of n. ve exactly n = 2k 2. Approximate the integral from Question 1 using integral, Matlab's built-in numerical integrator. What is the absolute error?

can i get some help with this ?

Answer 1

Matlab code for finding integration using different method clear all close all %function for which integration have to do f1-4 (x) exp (x) %Displaying the function for which integration have to do fprintf ( Function for which integration have to do f (x)) disp fl) tupper and lower limit a-0; b-1; %Function for exact integration value f_ext-exp (1)-exp (0) fprint f (Exact integration value for the function %f\n', fext) %loop for all integration values for i= 1:20 n 2 i+1; val_simp-Simpson (fl,a, b, n)i val_trap-Trapizoidal(f1,a,b,n) val_mid-Midpoint (fl,a, b, n)i error for all method errsimp (i)-abs(f_ext-val_simp); err-trap ( í )=abs ( f-ext-val-trap ) ; err_mid(i)-abs (f_ext-val mid); end ⅝Log log plot for error vs. step size loglog (k,err_simp) hold on loglog (k,err_trap) loglog (k,err_mid) legend'Simpson method','Trapizoidal method', "Mid point') xlabelNumber of steps') ylabelError of integration) title 'log plot of error vs steps using different method') integral value using matlab built in integral val_int-integral(fl,a,b); error in integration fprintf( "Error in integration for matlab built in integral is %e. n,abs(val_int-f_ext)) 888888888888888888888888888888888888888888钐888888888888888888888888 % Matlab function for mid point integration function val=Midpoint( func,a,b,N) % func is the function for integration % a is the lower limit of integration % b is the upper limit of integration % N number of rectangles to be used

val=0; splits interval a to b into N+1 xx-linspace (a, b, N+1); dx=xx(2)-xx (1); %x interval loop for Riemann integration subintervals for i-1: length ( xx )-1 val-val+dx double (func (xxl)) end %%Matlab function for mid point integration function val-Trapizoidal (func,a,b, N) % func is the function for integration % a is the lower limit of integration % b is the upper limit of integration % N number of rectangles to be used val 0; %splits interval a to b into N+1 subintervals xx-linspace (a,b, N+1) x-xx ( 2 )-xx ( 1 ) ; %x interval %loop for Riemann integration for i-2:length(xx)-1 val-val+dx*double (func (xx1)) end val-val+dx* (0.5*double ( func (xx(1))) +0.5*double(func (xx(end) ))); end %%Matlab function for mid point integration function val-simpson (func,a, b, N) % func is the function for integration a is the lower limit of integration % b is the upper limit of integration N number of rectangles to be used splits interval a to b into N+1 subintervals xx-linspace (a, b, N+1); dx-xx ( 2 )-xx ( 1 ) ; %x interval val-(dx/3) ( double (func(xx (1)))+double (func (xx (end)))) %loop for Riemann integration for i-2:length (xx)-1 if mod(i,2)0 val-val+ (dx/3) *4 double(func (xx1)); else val-val+ (dx/3) *2 double(func (xx1)); end end end 888888888888888888 End of Code 8888888888888888

Function for which integration have to do f(x)-(x)exp (x) Exact integration vlue for the function 1.718282 Error in integration for matlab built in integral is 4.440892e-16. log plot of error vs steps using different method Simpson method Trapizoidal method Mid point 10 10 10-10 12 10 14 10 10 10 Number of steps Published with MATLAB® R2018a

%Matlab code for finding integration using different method
clear all
close all
%function for which integration have to do
f1=@(x) exp(x);
%Displaying the function for which integration have to do
fprintf('Function for which integration have to do f(x)=')
disp(f1)

%upper and lower limit
a=0; b=1;

%Function for exact integration value
f_ext=exp(1)-exp(0);
fprintf('Exact integration value for the function %f\n',f_ext)

%loop for all integration values
for i=1:20
    n=2^i+1;
    val_simp=Simpson(f1,a,b,n);
    val_trap=Trapizoidal(f1,a,b,n);
    val_mid=Midpoint(f1,a,b,n);
    %error for all method
    err_simp(i)=abs(f_ext-val_simp);
    err_trap(i)=abs(f_ext-val_trap);
    err_mid(i)=abs(f_ext-val_mid);
    k(i)=n;
end

%Log log plot for error vs. step size
loglog(k,err_simp)
hold on
loglog(k,err_trap)
loglog(k,err_mid)
legend('Simpson method','Trapizoidal method','Mid point')
xlabel('Number of steps')
ylabel('Error of integration')
title('log plot of error vs steps using different method')

%integral value using matlab built in integral
val_int=integral(f1,a,b);
%error in integration
fprintf('Error in integration for matlab built in integral is %e.\n',abs(val_int-f_ext))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Matlab function for mid point integration
function val=Midpoint(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    val=0;
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    %loop for Riemann integration
        for i=1:length(xx)-1
            xx1=(xx(i)+xx(i+1))/2;
            val=val+dx*double(func(xx1));
        end
     
end

%%Matlab function for mid point integration
function val=Trapizoidal(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    val=0;
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            val=val+dx*double(func(xx1));
        end
       val=val+dx*(0.5*double(func(xx(1)))+0.5*double(func(xx(end))));
end

%%Matlab function for mid point integration
function val=Simpson(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    val=(dx/3)*(double(func(xx(1)))+double(func(xx(end))));
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            if mod(i,2)==0
                val=val+(dx/3)*4*double(func(xx1));
            else
                val=val+(dx/3)*2*double(func(xx1));
              
            end
        end
     
end

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