Suppose I have a function $f(\mathbf{x})$ across the surface of the unit hypersphere, where $\mathbf{x}=(x_1,...,x_d)'$ are the hyperspherical coordinates of a point on the surface of the unit hypersphere. I want to integrate this function over the positive orthant $\mathcal{S}^{p-1}_+$: $I=\int_{\mathcal{S}^{p-1}_+}f(\mathbf{x})d\mathbf{x}$. Let's assume this integral is analytically intractable, so I need to use numerical techniques, and $d$ is large, so Monte Carlo integration is more attractive that numerical quadrature approaches. Thus, I want to approximate the integral $I$ using Monte Carlo integration.
From Wikipedia, I will first sample $N$ points uniformly on the surface of the positive orthant. I am doing this by normalizing a vector of independent standard normal samples, then taking the absolute value of the normalized $d$-dimensional vector; this produces a uniform sample in $\mathcal{S}^{p-1}_+$. Then, I evaluate $f(\cdot)$ at each of these random samples and sum all the output together. Finally, I would need to multiple the sum by $V$ and divide by $N$, where $V$ is the volume of $\mathcal{S}^{p-1}_+$.
If everything looks right so far, then I just have one question: what is $V$? Is it the volume of the positive orthant, or the volume of the surface of the positive orthant? Technically, the point $\mathbf{x}$ can only lay on the surface of the sphere, so I'm tempted to thing $V$ is the "volume" (area) of the positive orthant, but I'm not too sure.
