How to plot function & integral with vector variable?

Question

I want to plot functions & integrals with vector variable the present in this paper.

For example the equation (5) at p. 3: I want to plot a chart like figure 5, i.e. the $x$-axis is the angle between vector $\mathbf S$ and $\mathbf V$ of equation (5), and the $y$-axis is the function value, and I also need to set condition that if $\mathbf R_l\cdot\mathbf V<0$, then the function value is $0$.

And a more complex one is the $H_s(\mathbf S)$ function at p. 8; this is an integral with vector variable and the integral domain is a hemisphere. How do I plot a chart also like figure 5?

Thanks.