Calculate the cumulative integral of a vector where the spacing between data points is 1.

Create a numeric vector of data.

Y = [1 4 9 16 25];

Y contains function values for $\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^2$ in the domain [1 5].

Use cumtrapz to integrate the data with unit spacing.

Q = cumtrapz(Y)

Q = 1×5

         0    2.5000    9.0000   21.5000   42.0000

This approximate integration yields a final value of 42. In this case, the exact answer is a little less, $41\frac{1}{3}$. The cumtrapz function overestimates the value of the integral because f(x) is concave up.

Calculate the cumulative integral of a vector where the spacing between data points is uniform, but not equal to 1.

Create a domain vector.

X = 0:pi/5:pi;

Calculate the sine of X.

Y = sin(X');

Cumulatively integrate Y using cumtrapz. When the spacing between points is constant, but not equal to 1, an alternative to creating a vector for X is to specify the scalar spacing value. In that case, cumtrapz(pi/5,Y) is the same as pi/5*cumtrapz(Y).

Q = cumtrapz(X,Y)

Q = 6×1

         0
    0.1847
    0.6681
    1.2657
    1.7491
    1.9338

Cumulatively integrate the rows of a matrix where the data has a nonuniform spacing.

Create a vector of x-coordinates and a matrix of observations that take place at the irregular intervals. The rows of Y represent velocity data, taken at the times contained in X, for three different trials.

X = [1 2.5 7 10];
Y = [5.2   7.7   9.6   13.2;
     4.8   7.0  10.5   14.5;
     4.9   6.5  10.2   13.8];

Use cumtrapz to integrate each row independently and find the cumulative distance traveled in each trial. Since the data is not evaluated at constant intervals, specify X to indicate the spacing between the data points. Specify dim = 2 since the data is in the rows of Y.

Q1 = cumtrapz(X,Y,2)

Q1 = 3×4

         0    9.6750   48.6000   82.8000
         0    8.8500   48.2250   85.7250
         0    8.5500   46.1250   82.1250

The result is a matrix of the same size as Y with the cumulative integral of each row.

Perform nested integrations in the x and y directions. Plot the results to visualize the cumulative integral value in both directions.

Create a grid of values for the domain.

x = -2:0.1:2;
y = -2:0.2:2;
[X,Y] = meshgrid(x,y);

Calculate the function $\mathit{f}\left(\mathit{x},\mathit{y}\right)=10{\mathit{x}}^2+20{\mathit{y}}^2$ on the grid.

F = 10*X.^2 + 20*Y.^2;

cumtrapz integrates numeric data rather than functional expressions, so in general the underlying function does not need to be known to use cumtrapz on a matrix of data. In cases where the functional expression is known, you can instead use integral, integral2, or integral3.

Use cumtrapz to approximate the double integral

To perform this double integration, use nested function calls to cumtrapz. The inner call first integrates the rows of data, then the outer call integrates the columns.

I = cumtrapz(y,cumtrapz(x,F,2));

Plot the surface representing the original function as well as the surface representing the cumulative integration. Each point on the surface of the cumulative integration gives an intermediate value of the double integral. The last value in I gives the overall approximation of the double integral, I(end) = 642.4. Mark this point in the plot with a red star.

surf(X,Y,F,'EdgeColor','none')
xlabel('X')
ylabel('Y')
hold on
surf(X,Y,I,'FaceAlpha',0.5,'EdgeColor','none')
plot3(X(end),Y(end),I(end),'r*')
hold off