MATLAB: Integration numerical with variable limits

integral2integrationnumerical integration

I am trying to find a double integration numerically where the inner integral has variable limits while the outer integral has scalar limits and not the other way round. As I understand integral2 allows you do the reverse of what I want i.e you can have the outer variable limits non scalar while the inner should be scalar.

    fxy=@(x,y)1/x+1/y
    xmin=0;
    xmax=@(y)2*y;
    ymin=0;
    ymax=+inf;
    integral2(fxy,xmin,xmax,ymin,ymax)
I get the following error 
Error using integral2 (line 76)
XMAX must be a floating point scalar.

Best Answer

The INTEGRAL2 interface is set up to calculate an iterated integral where the integral over the second argument of the integrand function f is the inner integral. It does not matter what you call your variables, whether it be x, y, t, p, r, or whatever. The first argument is the outer integral, and the second argument is the inner integral. If your problem happens to be formulated so that the inner integral variable is called x and the outer integral variable is called y, but your integrand is already defined so that x is the first argument and y is the second, then you just do this:

integral2(@(y,x)f(x,y),ymin,ymax,xmin,xmax)

Your example isn't integrable, or I'd demonstrate.

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MATLAB: How to perform double or triple integration in MATLAB

MATLAB provides several functions for the numerical evaluation of integrals. These functions are QUAD, QUADL, QUADV, DBLQUAD, and TRIPLEQUAD. Below is an example on how to implement a double integral using the DBLQUAD function. This shows how to solve the integral of f(x,y)dydx for the ranges of x going from xmin to xmax, and y going from ymin to ymax:

% This is the function to be integrated. 'y' is the inner
% variable and 'x' the outer variable.  Note the order that
% the parameters are passed into the anonymous function
% “f”: inner variable first, then outer variable. 
f = @(y,x) x*sin(y)+y*cos(x)
ymin = pi;
ymax = 2*pi;
xmin = 0;
xmax = pi;
result = dblquad(f, ymin, ymax, xmin, xmax);

The result is -9.8698, which is the correct result.

The first step is to build the function to be evaluated. The function must return a vector output when vector input is passed to it. You must also consider which variable is in the inner integral, and which belongs in the outer integral. In our example, the inner variable is y and the outer variable is x (the order in the integral is dydx).

The first argument is the handle of the integrand function. If you have defined this function as an anonymous function, no “@” symbol is required. The next four input arguments specify the limits of integration, and are:

 ymin       lower limit of inner integral             
 ymax      upper limit of the inner integral         
 xmin       lower limit of outer integral             
 xmax      upper limit of the outer integral

Note that if you are using a version of MATLAB older than MATLAB 7.0 (R14), you will need to use a function file or an inline function. To use a function file, try the following:

function out = f(y, x)
% The first (inner) variable, y, is supplied directly by 
% the 'quad' function.  We pass x using the mechanism for
% supplying extra parameters to the integrand function.
out = x*sin(y)+y*cos(x);  % The function to be integrated

Then, the integral can be evaluated using:

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

Also, note that the QUADV function was introduced in MATLAB 7.0 (R14), so it will not be available if you are using an older version.

MATLAB: How to plot a box on an existing plot

xmin=0;
xmax=1;
ymin=0;
ymax=1;
plot([xmin xmax xmax xmin xmin],[ymin ymin ymax ymax ymin])

rectangle('position',[xmin ymin xmin xmax])

Best Answer

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MATLAB: How to perform double or triple integration in MATLAB

MATLAB: How to plot a box on an existing plot

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