trapz

Calculate the 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 $$f(x)=x^2$$ in the domain [1, 5].

Use trapz to integrate the data with unit spacing.

Q = trapz(Y)

Q = 
42

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

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

Create a domain vector.

X = 0:pi/100:pi;

Calculate the sine of X.

Y = sin(X);

Integrate Y using trapz.

Q = trapz(X,Y)

Q = 
1.9998

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, trapz(pi/100,Y) is the same as pi/100*trapz(Y).

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 trapz to integrate each row independently and find the total 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 = trapz(X,Y,2)

Q1 = 3×1

   82.8000
   85.7250
   82.1250

The result is a column vector of integration values, one for each row in Y.

Create a grid of domain values.

x = -3:.1:3; 
y = -5:.1:5; 
[X,Y] = meshgrid(x,y);

Calculate the function $$f(x,y)=x^2+y^2$$ on the grid.

F = X.^2 + Y.^2;

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

Use trapz to approximate the double integral

To perform double or triple integrations on an array of numeric data, nest function calls to trapz.

I = trapz(y,trapz(x,F,2))

I = 
680.2000

trapz performs the integration over x first, producing a column vector. Then, the integration over y reduces the column vector to a single scalar. trapz slightly overestimates the exact answer of 680 because f(x,y) is concave up.