How to integrate a data set?

I have a data set that includes data for temperature and specific heat capacity, and I need to integrate (Cv(T)/T)dT and (Cv(T))dT over 0 to 270. I uploaded the data into Matlab already, and I have a plot of the data (temp on x-axis, Cv on y-axis). How do I code for the cumulative entropy (see first equation) and energy (see second equation)?

Answer 1

[Star Strider]

I would use the trapz function, or cumtrapz (linked to in the trapz documentation), depending on the result you want.

For example:

IntEq1 = trapz(T, Cv./T);
IntEq2 = trapz(T,Cv);

Without your data I am only guessing here, however these should at least be close to what you want. If you want vectors of interim results as well, replace trapz with cumtrapz in these assignments.

[Chuqi Zheng]

My data is a .csv file. Right now I have the data uploaded and in matrix form with

T = readtable(silver_data.csv);

C = T{:,:};

plot(C(:,1),C(:,2)).

That just gives me a plot of the data. Can I just directly add intEq1 after that?

[Star Strider]

I would do sometrhing like this:

SD = readtable('silver_data.csv', 'PreserveVariableNames',1);

T = SD{:,1};
Cv = SD{:,2};

IntEq1 = trapz(T, Cv./T)
IntEq2 = trapz(T,Cv)

figure
plot(T, Cv)
grid

producing:

IntEq1 =
          9.70376967628969

IntEq2 =
          1205.67613588139

[Chuqi Zheng]

Thank you!! I’ll try that. My teacher mentioned using Inte=cumtrapz though? Because I’ll need multiple cumulative entropies and energies (one for each time entry)?

[Star Strider]

My pleasure.

As I mentioned in my original Answer:

‘If you want vectors of interim results as well, replace trapz with cumtrapz in these assignments.’

The code then becomes:

SD = readtable('silver_data.csv', 'PreserveVariableNames',1);
T = SD{:,1};
Cv = SD{:,2};
vars = SD.Properties.VariableNames;

IntEq1 = cumtrapz(T, Cv./T);
IntEq2 = cumtrapz(T,Cv);

figure
yyaxis left
plot(T, Cv)
grid
xlabel(vars{1})
ylabel(vars{2})

yyaxis right
hyr1 = semilogy(T, IntEq1);
hold on
hyr2 = plot(T, IntEq2);
hold off
grid
ylabel('Values of Integrals')
legend([hyr1 hyr2],'IntEq1', 'IntEq2', 'Location','SE')

This calculates them and plots them. I use the semilogy scale on the right axis so that both curves are easily visible.

[Chuqi Zheng]

Got it, thank you! Will this show me a table with the values as well or just the graph?

[Star Strider]

My pleasure.

That code will show you the plot.

This assignment:

SD = [SD table(IntEq1, IntEq2)];

will add them to the ‘SD’ table, so:

FirstFiveRows = SD(1:5,:)

produces:

FirstFiveRows =
    Temp(K)    C_v cal/deg/mole     IntEq1     IntEq2 
    _______    ________________    ________    _______

    9.8181         0.042043               0          0
    11.786         0.078029        0.010728    0.11815
    13.534          0.11397        0.023877      0.286
     14.84          0.16766        0.036748    0.46983
    16.367          0.21248        0.055285    0.76005

In the code, put that just after you calculate the integrals:

IntEq1 = cumtrapz(T, Cv./T);
IntEq2 = cumtrapz(T,Cv);
SD = [SD table(IntEq1, IntEq2)];

[Chuqi Zheng]

Thank you again, you're a lifesaver. Should I put the code for the table values in after the code for the plot?

[Star Strider]

My pleasure.

So long as you put the plots and the table concatenation code after the integration assignments, you can put them anywhere in your code you want. That is entirely your decision.

If my Answer helped you solve your problem, please Accept it!

[Chuqi Zheng]

Thank you so so so much. Sorry to ask this, but I'm completely new to coding, so would you mind explaining what some of your steps mean? Like what does SD mean? The hyr's? vars1?

[Star Strider]

Sure!

Your file is called ‘silver_data.csv’ so I called the table it created ‘SD’. (I might have called it ‘T’ for ‘table’, however you have a variable named ‘T’ for temperature.)

The ‘hyr’ variables are handles to the particular yyaxis right plots. They guarantee that the legend only refers to the curves in yyaxis right, and do not include the yyaxis left variables. I could have instead created it as:

legend('Data', 'IntEq1', 'IntEq2', 'Location','SE')

to include all of them.

The ‘vars’ variable contains the column headers of the ‘SD’ table, imported from the ‘silver_data.csv’ file. I used it to create the ylabel for yyaxis left and the common xlabel.

[Chuqi Zheng]

Got it! Thanks again for all your help. I really appreciate it.

[Star Strider]

As always, my pleasure!