representing simple definite integral as convolution in matlab using CONV

Question

A definite integral is a convolution with idial all '1' filter(by the theory) so i tried to test this issue with MATLAB. i have taken a function called fun made an integral over it in 1e-6 - 20*e-6 interval the result is called simbolic_int_result which 37.7 then i have taken fun sampled it in 1e-6 - 20*e-6 interval as shown bellow with 1000 samples and created all "1" filter (idial) and used CONV to make convolution,but its not working, instead of a scalar i get 2000X1 vector Where did i go wrong? Thanks.

c0=2.997*10.^8;% m/s speed of light in vaccum
h=6.625*10.^-34;% J.s Planck constant
k=1.38*10.^-23;%  T/K Boltzmann constant
n=1;

T=307;%  Temperature initiatlisation
fun=@(lambda)(2*pi.*h.*(c0.^2))./((n.^2).*(lambda.^5).*(exp((h.*c0)./(k.*T.*lambda))-1));
simbolic_int_result= integral(fun,1e-6,20e-6)/10;

k=linspace(10.^-6,20*10^-6,1000);
fun_vector=fun(k);

filter=linspace(1,1,1000); % all ones idial filter

convolution_result=conv(filter,fun_vector);

Answer 1

A definite integral is a convolution with idial all '1' filter(by the theory)

No. Where does that theory come from ?

If you want to approximate a definite integral with a discrete sequence, you can simply sum up the samples in the integration interval and scale the sum by the sampling interval.

The output of a convolution of two functions is another function, not a single value.