Fail time with joint density $f_{X,Y}(x,y)=6e^{-x}e^{-2y}, \quad 0<x<y< \infty$ The following is the problem I am solving for.

Let $X$ and $Y$ be the fail time for a machine with two components.  The component $Y$ will start working if and only if component $X$ fails.  The machine fails when component $Y$ fails as well.  Given the joint density function
$$f_{X,Y}(x,y)=6e^{-x}e^{-2y}, \quad 0&lt;x&lt;y&lt; \infty$$
find the expected time of failure of the machine.

My method was to find
$$\int_0^{\infty} \int_x^{\infty} (x+y)6e^{-x}e^{-2y} dydx$$
which ended up being equal to $ 7 \over 6$.
However, the answer was supposedly approximately 0.83.
I am not quite sure how to solve this problem.
Can someone explain me what is going on?
Hint:

The answer is not approximately 0.83, but $\frac{5}{6}$.  Use the following formulation  $$\int_{0}^{\infty} \int_{0}^{y} (y)(6e^{-x}e^{-2y})dxdy$$.  A tedious integration by parts and you will land up with the answer $\frac{5}{6}$
Hint:

The answer is not approximately 0.83, but $\frac{5}{6}$.  Use the following formulation  $$\int_{0}^{\infty} \int_{0}^{y} (y)(6e^{-x}e^{-2y})dxdy$$.  A tedious integration by parts and you will land up with the answer $\frac{5}{6}$