Trapezoidal numerical integration
Y is a vector, then
the approximate integral of
Y is a matrix, then
over each column and returns a row vector of integration values.
Y is a multidimensional array,
trapz(Y) integrates over the first dimension
whose size does not equal 1. The size of this dimension becomes 1,
and the sizes of other dimensions remain unchanged.
Q = trapz(___, integrates along the
dim using any of the previous syntaxes. You must
Y, and optionally can specify
If you specify
X, then it can be a scalar or a vector with
length equal to
size(Y,dim). For example, if
Y is a matrix, then
integrates each row of
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 in the domain [1, 5].
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, . 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
Y = sin(X);
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
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];
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
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
Create a grid of domain values.
x = -3:.1:3; y = -5:.1:5; [X,Y] = meshgrid(x,y);
Calculate the function 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
trapz to approximate the double integral
To perform double or triple integrations on an array of numeric data, nest function calls to
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.
Y— Numeric data
Numeric data, specified as a vector, matrix, or multidimensional
array. By default,
trapz integrates along the first
Y whose size does not equal 1.
Complex Number Support: Yes
X— Point spacing
1(default) | uniform scalar spacing | vector of coordinates
Point spacing, specified as
1 (default), a uniform
scalar spacing, or a vector of coordinates.
X is a scalar, then it specifies a uniform
spacing between the data points and
X is a vector, then it specifies
x-coordinates for the data points and
length(X) must be the same as the size of the
integration dimension in
dim— Dimension to operate along
Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.
Consider a two-dimensional input array,
trapz(Y,1) works on successive elements in the columns of
Y and returns a row vector.
trapz(Y,2) works on successive elements in the rows of
and returns a column vector.
dim is greater than
trapz returns an array of zeros of the same
trapz performs numerical integration via the trapezoidal
method. This method approximates the integration over an interval by breaking the
area down into trapezoids with more easily computable areas. For example, here is a
trapezoidal integration of the sine function using eight evenly-spaced
For an integration with
N+1 evenly spaced
points, the approximation is
where the spacing between each point is equal to the scalar value . By default MATLAB® uses a spacing of 1.
If the spacing between the
N+1 points is not constant, then the formula
where , and is the spacing between each consecutive pair of points.
perform numerical integrations on discrete data sets. Use
integral3 instead if a functional expression
for the data is available.
trapz reduces the size of the dimension
it operates on to 1, and returns only the final integration value.
returns the intermediate integration values, preserving the size of
the dimension it operates on.
Usage notes and limitations:
If you supply
dim, then it must
be a constant.
See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).