You need to numerically integrate a double integral (for example, to compute the volume of some arbitrary shape or the volume under a surface).
Separate the problem into two parts, each involving only single integrals, and then apply the numerical integration techniques discussed earlier in this chapter.
Computing multiple integrals numerically can be challenging, especially when triple or higher integrals and involved. To the best of my knowledge, there are no general-purpose numerical integration formulas that handle multiple integrals. There is a method called Monte Carlo integration that involves taking a lot of random samples within the volume or domain being integrated and using the distribution of those samples to estimate the volume or whatever is of interest in higher dimensions. Alternatively, you can break up a multiple integral into successive single integrals and apply the standard numerical integration techniques discussed in this chapter.
To demonstrate this latter approach, consider the surface shown in Figure 10-3.
Figure 10-3. 3D surface
Let's assume you want to compute the volume under that surface bounded by y = [0,1] and x = [0,1]. This is a three-dimensional problem, but instead of computing a multiple integral, you can compute a series of single integrals.
The first step is to consider cross-sections of ...