MATLAB®

Integration

Introduction

The area beneath a section of a function F(x) can be determined by numerically integrating F(x), a process referred to as quadrature. The MATLAB^® quadrature functions are:

To integrate the function defined by humps.m from 0 to 1, use

q = quad(@humps,0,1)

q =
    29.8583

Both quad and quadl operate recursively. If either method detects a possible singularity, it prints a warning.

You can include a fourth argument for quad or quadl that specifies an absolute error tolerance for the integration. If a nonzero fifth argument is passed to quad or quadl, the function evaluations are traced.

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Example: Arc Length

You can use quad or quadl to compute the length of a curve. Consider the curve parameterized by the equations

where .

A three-dimensional plot of this curve is

t = 0:0.1:3*pi;
plot3(sin(2*t),cos(t),t)

The arc length formula says the length of the curve is the integral of the norm of the derivatives of the parameterized equations

The function hcurve computes the integrand

function f = hcurve(t)
f = sqrt(4*cos(2*t).^2 + sin(t).^2 + 1);

Integrate this function with a call to quad

len = quad(@hcurve,0,3*pi)

len =
   1.7222e+01

The length of this curve is about 17.2.

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Example: Double Integration

Consider the numerical solution of

For this example f(x,y) = ysin(x) + xcos(y). The first step is to build the function to be evaluated. The function must be capable of returning a vector output when given a vector input. You must also consider which variable is in the inner integral, and which goes in the outer integral. In this example, the inner variable is x and the outer variable is y (the order in the integral is dxdy). In this case, the integrand function is

function out = integrnd(x,y)
out = y*sin(x) + x*cos(y); 

To perform the integration, two functions are available in the funfun directory. The first, dblquad, is called directly from the command line. This M-file evaluates the outer loop using quad. At each iteration, quad calls the second helper function that evaluates the inner loop.

To evaluate the double integral, use

result = dblquad(@integrnd,xmin,xmax,ymin,ymax);

The first argument is a string with the name of the integrand function. The second to fifth arguments are

Here is a numeric example that illustrates the use of dblquad.

xmin = pi;
xmax = 2*pi;
ymin = 0;
ymax = pi;
result = dblquad(@integrnd,xmin,xmax,ymin,ymax)

The result is -9.8698.

By default, dblquad calls quad. To integrate the previous example using quadl (with the default values for the tolerance argument), use

result = dblquad(@integrnd,xmin,xmax,ymin,ymax,[],@quadl);

Alternatively, you can pass any user-defined quadrature function name to dblquad as long as the quadrature function has the same calling and return arguments as quad.

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Optimization

Differential Equations