probability - an problem for the range of integration of cdf

Let X be a continuous, nonnegative random variable, and $F_X(x)$ is the cdf of X.

$\int_0^\infty P(X>x)dx=\int_0^\infty\int_x^\infty f_x(y)dydx=\int_0^\infty\int_0^ydxf_x(y)dy$

I don't know why the range is from $0$ to $y$ at the last term.

Thanks.

Answer

Draw the line $y=x$. In the middle integral, $y$ went from $y=x$ to $\infty$, and then $x$ went from $x=0$ to $\infty$.

So the full double integration was done over the part $K$ of the first quadrant that is above the line $y=x$. Concentrate on $K$, a roughly triangular region. Write $K$ in that reg. on.

Now we change the order of integration. For any fixed $y$, we integrate first with respect to $x$. So $x$ goes from $x=0$ to the diagonal line. We have to stop there in order not to leave $K$. Thus $x$ goes from $x=0$ to $x=y$. Then $y$ goes from $0$ to $\infty$.

Remark: The rest of the story is interesting. When we integrate with respect to $x$, where $x$ goes from $0$ to $y$, then $f_X(y)$ is being treated as a constant, and we get $yf_X(y)$. Now when we integrate with respect to $y$, we get $E(X)$.