probability - Intuitive explanation for $mathbb{E}X= int_0^infty 1-F(x) , dx$

I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that is true? If at each point we sum the probability $\mathbb{P}(X>x)$, why should we end up with the expectation?

Thanks