# Numerical integration of a 2D hemisphere discrete dataset where limits are unknown

I am trying to compute the integral of a 2D hemisphere dataset $$f_r \, (\theta, \phi)$$ where $$\theta \in [0, \pi / 2[$$ and $$\phi \in [0, 2\pi]$$. I am making the measurements myself, so I can choose the number of values and their coordinates.

My first idea was to use the different points to triangulate my definition set, and sum the mean value of the vertices of each triangle divided by their surface.

But my problem will appear when $$\theta \rightarrow \pi/2$$, because I cannot have any measurement for these values. Is there a good strategy to solve this problem ?

• Have you taken a look to Lebedev quadrature? Jun 23 at 13:10
• Is your problem computing the integral without evaluating at $\theta=\pi/2$? If that's the case, not knowing that particular value should not affect the result since it is a set of measure 0. Also, it is somewhat arbitrary where $\theta=\pi/2$ is located in a sphere. Jun 23 at 13:20