I am not able to integrate heaviside(y-f) for x= 0 to pi/2 and y = 0 to pi/2 (only syms integration "int"). After running the code, output is shown in below and i am not able to get the numerical answer. Can you give some ideas on how to proceed?

syms x y
f = sin(x);
integ = int( int(heaviside(y-f), x, 0, pi/2),y, 0, pi/2)

Result:

integ =

int(int(heaviside(y - sin(x)), x, 0, pi/2), y, 0, pi/2)

[Walter Roberson]

Formally speaking, Heaviside is undefined when the expression value is exactly 0. There are multiple conventions about what value is to be substituted in that case. Which convention do you want to use?

[CHINTALAPUDI GNANA SURYA NARAYANA SAI]

Actually i don't know what do you mean by convention?? and thank you very much for answering it by "Reverse the order of integration".

[Walter Roberson]

For your purposes, is heaviside(0) equal to 0, or is it equal to 1, or is it equal to 1/2, or is it undefined ?

[CHINTALAPUDI GNANA SURYA NARAYANA SAI]

yes, it is heaviside(=0 or >0) equal to 1 and heaviside(<0) equal to 0.

Answer 1

[Walter Roberson]

Reverse the order of integration.

>> integ = int( int(heaviside(y-f), y, 0, pi/2),x, 0, pi/2)
 
integ =
 
pi^2/4 - 1

Answer 2

[hosein Javan]

if you need numerical integration, some functions cannot be easily evaluated by symbolic. instead use "integral". in here we have a double integral. the integral calculation method is 'iterated' which means on each iteration it is integrating on one dimension and then the next. it takes a few seconds to accomplish though.

fun = @(x,y) double( y > sin(x) )
q = integral2(fun, 0, pi/2, 0, pi/2, 'Method', 'iterated', 'Reltol', 1e-8)
% answer in command window:
q =
    1.4674

here's your function plot. blue part is zero. yellow part equals one.

[hosein Javan]

notice that because one side is totally zero (where y<sin(x)), we can perform the integral by defining the lower limit of the integral in terms of a variable. this time notice that it has become much faster and accurate.

while before it took about 1.2 seconds.

fun = @(x,y) ones(size(x))
tic
q = integral2(fun, 0, pi/2, @(x) sin(x), pi/2, 'RelTol',eps)
toc

q =

1.4674

Elapsed time is 0.005605 seconds.

[CHINTALAPUDI GNANA SURYA NARAYANA SAI]

actually i need to do syms int type of integration, but as you suggested i will also look into integral2 command. Thank you very much.

[hosein Javan]

you're welcome.