The integral of the function $f(x)=1/x^2$ is convergent and it equals 1 when the limits of the integral is $\int_1^\infty$ but it's divergent and equals $\infty$ when the limits are $\int_0^1$.
I know the math but I want to understand the reason intuitively(in layman's terms). Both of this function's parts look similar to each other(I know they are NOT identical) so Why don't they integrate to a similar value?.
Another example: the integral of the normal distribution is $1$ but the integral of the beta function(with $\alpha$ and $\beta$ equal $0$) $B(0,0)$ is $\infty$. Either x or y axis goes to infinity in both of those functions so Why are their integrals different?