Using Simpson's rule to integrate function

I have a function in which I'm trying to solve by integrating. I must use Simpson's rule with for loops to solve. This is what I have so far but I'm unsure how to tailor this better to what I need: I'm trying to integrate the "mass flow" function.

syms p T R A time
rho = 1.225; %kg/m^3
T = 288.15; %Kelvin
R = 287.057; %m^2/s^2*K
A = 0.01; %area of wind tunnel
gamma = 1.4;
M = 2.4; %Mach # that V0 is running at
a = sqrt(gamma*R*T);%speed of sound of air
V0 = M*a; %speed of V0
VF = 0.00001; %m/s

VelocityFunction = 0.00001 == 816.71*exp(-10*time);
TotalTime = solve(VelocityFunction, time);
TotalTime = double(TotalTime);

massflow = @(t) rho*A*V0*exp(-10*t);

delX = 0.01;
XVals = 0:delX:1;

fvals = f(XVals);
resultTrap = 0;
for idx = 1:length(XVals)-1
    resultTrap = resultTrap + (fvals(idx) + fvals(idx + 1))/2*delX;
end

myResult = resultTrap

xVals = 0;
for idx = 1:10   %first value will be zero
    xVals(idx+1) = xVals(idx) + rand(1);
end
xVals = xVals/(max(xVals)) %normalizing output here
fVals = f(xVals);
resultTrapRand = 0;
for idx = 1:length(xVals)-1
    resultTrapRand = resultTrapRand + (fVals(idx) + fVals(idx +1))/2*(xVals(idx+1) - xVals(idx));
end

myResultRand = resultTrapRand

[Jim Riggs]

I am rather confused by your code.

What function are you wanting to integrate?

You have defined a function "massflow", but this function is never referenced.

It looks like you are referencing some function "f" (as in f(XVals)) but f is not defined.

[Ryan Bowman]

This is actually I mistake I have made. I am wanting to integrate the "massflow" function.

Answer 1

[Jim Riggs]

OK. now I understand.

I prefer to code an integration loop as follows;

For the trapezoidal solution:

StartTime = 0;
EndTime = 1.0;
Tstep = 0.001;
Tsum = 0;
for time = StartTime+Tstep:Tstep:EndTime
    Tsum = Tsum + Tstep/2*(massflow(time-Tstep) + massflow(time));
end

Tsum is the Trapezoidal integration solution.

For the Simpson's rule solution:

Sstep = 0.02;
Ssum = 0;
for time = StartTime+2*Sstep:2*Sstep:EndTime
    Ssum = Ssum + Sstep/3*(massflow(time-2*Sstep) + 4*massflow(time-Sstep) + massflow(time));
end

Ssum is the Simpson's rule integration solution.

Using the above, you can set a different step size for the trapezoidal method and Simpson's method.

You will see that the Trapezoidal rule solution requires a step that is several orders of magnitude smaller than Simpson's rule for similar accuracy.