I want to find the probability of finding an oscillator between x and x+dx. I calculated the volume enclosed in the phase space for the oscillator with energy between E and E+dE $\frac{8\pi EdE}{\omega^2}$. This is the total volume $V_{T}$. Now, I want to get the volume in $[x,x+dx]\times[E,E+dE]$ $V$. And that should give me the probability of finding the oscillator between x and x+dx as $P=\frac{V}{V_{T}}$. However, I dont know how to calculate V, the only way I see is by integrating $\sqrt{1-x^2}$ which is obtained from the ellipse equation. But that integral gives me arcsines and it gets too complicated. I should get the answer in terms of the energy E and the position x.

To get $V_{T}$ I used the fact that the area of an ellipse is $\pi a b$ with a and b its semiaxis, and I didnt take into account the quadtratic term $dE^2$

I want to find the probability of finding an oscillator between $x$ and $x+dx$. I calculated the volume $\frac{8\pi EdE}{\omega^2}$ enclosed in the phase space for the oscillator with energy between $E$ and $E+dE $. This is the total volume $V_{T}$. Now, I want to get the volume in $[x,x+dx]\times[E,E+dE]$ $V$. And that should give me the probability of finding the oscillator between $x$ and $x+dx$ as $P=\frac{V}{V_{T}}$. However, I dont know how to calculate $V$, the only way I see is by integrating $\sqrt{1-x^2}$ which is obtained from the ellipse equation. But that integral gives me arcsines and it gets too complicated. I should get the answer in terms of the energy $E$ and the position $x$.

To get $V_{T}$ I used the fact that the area of an ellipse is $\pi a b$ with a and b its semiaxis, and I didnt take into account the quadtratic term $dE^2$