I don't understand this equation. Any help is appreciated.
equation $$ \underbrace{ \frac{N_{\rm hits}}{\rm trials}=\frac1N\sum_{i=1}^N{\cal O}_i }_{\rm sampling} \simeq \underbrace{ \langle{\cal O}\rangle = \frac{\int_{-1}^1dx\int_{-1}^1dy\,\pi(x,y)\,{\cal O}(x,y)}{\int_{-1}^1dx\int_{-1}^1dy\,\pi(x,y)} }_{\rm integration} $$
Also, which are the best sources to understand equations and notations in general.
there is an integral $dx$. should it not be integral $(f(x) dx)$ if you have nothing between integral symbol and $dx$ (not even a constant) what will it evaluate to??
Along with you down votes describe what needs to be done to improve the question. of this needs to move to another forum (if so why?)
Your confusion is about some sleek integration notation $\int dx\,f(x)$ that is mostly found in applied mathematics and physics. You have to interpret the expression and scan for all terms containing the integration variables to find what the integrand is. In proper notation where $\int$ and $dx$ bracket the integrand so that less guessing is involved, the last fraction should be $$ \frac{\int_{-1}^1\int_{-1}^1\pi(x,y)\,{\cal O}(x,y)\,dy\,dx}{\int_{-1}^1\int_{-1}^1\,\pi(x,y)\,dy\,dx} $$ which is the typical structure for an expectation where $\pi$ is some weight function that is not necessarily a probability density on the square $[-1,1]^2$. As $\cal O$ appears to be an indicator function (that is, taking only values $0$ and $1$) for the set $A=\{(x,y):{\cal O}(x,y)=1\}$, the quotient gives the fraction of the mass of $A$ relative to the mass of the full square.