Probability density function is $ y = 2x $ for $[0,1]$, it is zero anywhere else.
What is the expectation?
I think it is $E[X]=\int_0^1 x2x dx=\frac{2}{3}x^3|_0^1=\frac{2}{3}$
This is not correct, it should be $\frac{1}{\sqrt{2}}$.
Where did I go wrong?
$E[X] = \dfrac23 \approx 0.667$ as you have correctly calculated.
$\dfrac{1}{\sqrt{2}}\approx 0.707$ is in fact the median rather than the mean, in the sense that both $$\int_0^{\frac{1}{\sqrt{2}}} 2x \; dx = \frac12$$ and $$\int_{\frac{1}{\sqrt{2}}}^1 2x \; dx = \frac12.$$