# Issue with numerical integration of two variable function

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Bathala Teja on 20 Sep 2021
Commented: Bathala Teja on 22 Sep 2021
I want to do integration of two variable function w.r.t one variable.
A = @(x, y)cos(x)+sin(y)
A = function_handle with value:
@(x,y)cos(x)+sin(y)
B = @(x, y)(A-integral(@(x)A, 0 , 2*pi))
B = function_handle with value:
@(x,y)(A-integral(@(x)A,0,2*pi))
How to see the output here??

Walter Roberson on 20 Sep 2021
A = @(x, y) cos(x)+sin(y)
A = function_handle with value:
@(x,y)cos(x)+sin(y)
B = @(x, y) A(x,y)-integral(@(X)A(X,y), 0 , 2*pi, 'ArrayValued', true)
B = function_handle with value:
@(x,y)A(x,y)-integral(@(X)A(X,y),0,2*pi,'ArrayValued',true)
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.1e-07. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.1e-07. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
fsurf(B, [-pi pi -pi pi])
Walter Roberson on 22 Sep 2021
It depends what you mean by "possible".
If you mean that you want to get the formula cos(x) + sin(y) - 2 * pi * sin(y) as your output, then that is certainly not something you could ever get through numeric methods, so it would require either using the Symbolic Toolbox or else doing equivalent processing such as text manipulation.
You know from examination that cos(x) + sin(y) is separable, so int(cos(x) + sin(y), x, 0, 2*pi) can be expanded into int(cos(x),x, 0, 2*pi) + int(sin(y), x, 0, 2 * pi) .
The second of those is constant in x, so the second of those is sin(y) * (2*pi) - sin(y) * (0) which is sin(y) * 2*pi
The first part, int(cos(x), x, 0, 2*pi) is (-sin(x) evaluated at 2*pi) - (-sin(x) evaluated at 0) which is -0 - (-0) which is 0.
So the entire integral is going to come out as sin(y) * 2*pi
You then subtract that from the original function cos(x) + sin(y) so you are going to get cos(x) + sin(y) - sin(y) * 2*pi
This approach requires a small knowledge of the integral of trig functions -- knowing how to integrate cos(x).
But this approach requires human knowledge of how to read the formula cos(x) + sin(y) and the formula involving the integral. The parts of MATLAB that are able to read a formula like that and have the knowledge of what it means and how to integrate... well, that's exactly what the Symbolic Toolbox does.
So as a human, you can find the solution fairly easily. Biut as a computer, you need the Symbolic Toolbox, or at least software that is programmed with enough calculus knowledge to handle all of the cases you need.
Bathala Teja on 22 Sep 2021
ok got it
thank you

Sargondjani on 20 Sep 2021
Edited: Sargondjani on 20 Sep 2021
You created only function handles. So f(x,y) = .....
To compute the numerical value you need to assign numerical values to x and y. So for example:
y=0;
x=1;
B_value = B(x,y);

Steven Lord on 20 Sep 2021
A = @(x, y)cos(x)+sin(y);
B = @(x, y)(A-integral(@(x)A, 0 , 2*pi));
Your expression for B won't work for two reasons. The integral function requires the function handle you pass into it as the first input to return a numeric array, but yours returns a function handle. Even if that worked, you would then try to subtract that result from a function handle and arithmetic on function handles is not supported.
From your description it sounds like you want B to be a function of one variable, but you need to pass two inputs into your A function handle. Assuming you want to fix the value of x and make B just a function of y, evaluate A inside your expression for B:
A = @(x, y)cos(x)+sin(y);
fixedX = 0.5;
B = @(y) A(fixedX, y)-integral(@(z)A(fixedX, z), 0 , 2*pi)
B = function_handle with value:
@(y)A(fixedX,y)-integral(@(z)A(fixedX,z),0,2*pi)
B(0.25)
ans = -4.3890
Bathala Teja on 21 Sep 2021
My intension is to get B as a function of x and y not interms of value

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