Question

1.1) Write a script in MATLAB that uses Reimann sum with 100 intervals (n=100) to calculate hydrostatic force of water on a triangular gate submerged under water (Figure 1). Give two estimates of this force (lower and upper bound for value of force). Hint: You may need to first find the hydrostatic force on the thin hashed slice by hand using the following information. Hydrostatic pressure P = pgz. p = 10003,= 9.81 m2/s. Note that h = 7-z and dA-W*dz. Then integrate the force over all slices on the to find the total force exerted on the gate. Total hydrostatic force = S dF, dF=PdA. 1.2) 1.3) Repeat the Reiman sum, this time with n=500. Present left and right Reimann sum values of force. Trapezoidal integration: Compute the value of force (the integration) using the trapz function in MATLAB using n=100. 1.4) Compute the value of force this time using the symbolic toolbox of MATLAB (int function, lab7). 1.5) Setting symbolic analytical solution of integral as the actual value of this integral, compute the percent of error between all numerical methods and actual value as the following: Numerical methods that need to be compared: Reimann sum left and right for n=100 and n=500, trapz. Error = numerical method value - actual value * 100 actual value Water surface 3 m W=h/2 WWW dA=W*dz Gate z P=pgz

Answer 1

. P= pg2 Here 0 = 1000 kg g 29.81 m² is allow Force d F 2 Pd A . 2PX W dz h. z (1-2) .=P4 h dz. oPX (1-2 da 20g ZX (7-2) dz T2 Fej eg ZX (7-2) dz j1006 X 3.81 X 2 (7-2) dz 7 2 7

Matlab code for all types of integrations clear all close all %function for which integration have to do func=@(z) (1000*9.81.*2. *(7-2))/2; fprintf('function for which integration have to do f(2)=') disp(func) fall limits a=1; b=7; fprintf('Upper and lower limit of integral h=%f to b=%f\n', a,b) %number of subintervals N=100; fall integration value using different method vall=left_riemann(func, a,b,N); val2=right_riemann (func, a, b, N); fprintf('For N = %d\n',N) fprintf('\tIntegral value using left Reimann sum =%f\n', vall) fprintf('\tIntegral value using Right Reimann sum = f\n',val2) $number of subintervals N=500; fall integration value using different method vall=left_riemann(func, a,b,N); val2=right_riemann (func, a, b,N); fprintf('For N = %d\n',N) fprintf('\tIntegral value using left Reimann sum =%f\n', vall) fprintf('\tIntegral value using Right Reimann sum =%f\n',val 2) %integral using trapz zz=linspace(a, b, 100); ff=func(zz); val3=trapz (zz, ff); fprintf('For n=%d, integral using trapz =%f\n',100,val3) syms z f(2)=( 1000*9.81*2* (7-2))/2; int_zz (2)=int(f, z); fprintf('Integral using int function') disp(int_zz) ext_val=double(int_zz(b)-int_zz(a)); fprintf('Integral value for limit h=%f to b=%f is f\n', a,b, ext_val) Error values errorl=(((ext_val-vall))/ext_val)*100; error2=(((ext_val-val2))/ext_val)*100; error3=(((ext_val-va13))/ext_val)*100; %Printing the result fprintf('\tFor Left Reimann sum relative percent error is fe \n', errori) fprintf('\tFor Right Reimann sum relative percent error is fe \n', error2) fprintf('\tFor trapz relative percent error is %e\n',error3)

%% Matlab function for Left Riemann integration function val=left_riemann(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; Esplits 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); val=val+dx*double(func (xx1)); end end %% Matlab function for right Riemann integration function val=right_riemann(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; Esplits 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) xx1=xx (i); val=val+dx* double(func(xx1)); end end %-------------- End of Code -------------- @(2) function for which integration have to do f(2)= (1000*9.81.*2. *(7-2))/2 Upper and lower limit of integral h=1.000000 to b=7.000000 For N = 100 Integral value using left Reimann sum = 265735.242000 Integral value using Right Reimann sum =263969.442000 For N = 500 Integral value using left Reimann sum =265045.873680 Integral value using Right Reimann sum =264692.713680 For n=100, integral using trapz =264851.983471 Integral using int function -(1635+2^2* (2*2 - 21 ) )/2 symbolic function inputs: z Integral value for limit h=1.000000 to b=7.000000 is 264870.000000 For Left Reimann sum relative percent error is -6.640000e-02 For Right Reimann sum relative percent error is 6.693333e-02 For trapz relative percent error is 6.802027e-03

%Matlab code for all types of integrations
clear all
close all
%function for which integration have to do
func=@(z) (1000*9.81.*z.*(7-z))/2;
fprintf('function for which integration have to do f(z)=')
disp(func)
%all limits
a=1; b=7;
fprintf('Upper and lower limit of integral h=%f to b=%f\n',a,b)
%number of subintervals
N=100;
%all integration value using different method
val1=left_riemann(func,a,b,N);
val2=right_riemann(func,a,b,N);
fprintf('For N = %d\n',N)
fprintf('\tIntegral value using left Reimann sum =%f\n',val1)
fprintf('\tIntegral value using Right Reimann sum =%f\n',val2)
%number of subintervals
N=500;
%all integration value using different method
val1=left_riemann(func,a,b,N);
val2=right_riemann(func,a,b,N);
fprintf('For N = %d\n',N)
fprintf('\tIntegral value using left Reimann sum =%f\n',val1)
fprintf('\tIntegral value using Right Reimann sum =%f\n',val2)

%integral using trapz
zz=linspace(a,b,100);
ff=func(zz);
val3=trapz(zz,ff);
fprintf('For n=%d, integral using trapz =%f\n',100,val3)

syms z
f(z)=(1000*9.81*z*(7-z))/2;
int_zz(z)=int(f,z);
fprintf('Integral using int function ')
disp(int_zz)
ext_val=double(int_zz(b)-int_zz(a));
fprintf('Integral value for limit h=%f to b=%f is %f\n',a,b,ext_val)

%Error values
error1=(((ext_val-val1))/ext_val)*100;
error2=(((ext_val-val2))/ext_val)*100;
error3=(((ext_val-val3))/ext_val)*100;

%Printing the result
fprintf('\tFor Left Reimann sum relative percent error is %e\n',error1)
fprintf('\tFor Right Reimann sum relative percent error is %e\n',error2)
fprintf('\tFor trapz relative percent error is %e\n',error3)

%%Matlab function for Left Riemann integration
function val=left_riemann(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);
            val=val+dx*double(func(xx1));
        end   
end
%%Matlab function for right Riemann integration
function val=right_riemann(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)
            xx1=xx(i);
            val=val+dx*double(func(xx1));
        end   
end
%-------------- End of Code --------------%