Final answer:
Both the trapezoidal and Simpson's rule can be used to evaluate double integrals numerically. The trapezoidal rule uses linear segments, while Simpson's rule uses parabolic segments for approximation. Numerical methods are particularly useful when the integral cannot be solved analytically or when dealing with tabulated integrands.
Step-by-step explanation:
The question deals with the application of numerical methods, specifically the trapezoidal and Simpson’s rule, for the evaluation of double integrals. The correct answer to whether only the trapezoidal rule, only Simpson's rule, both rules, or neither of them are applicable is:
C. Both Trapezoidal and Simpson’s rules are applicable
These rules can be extended to compute double integrals by applying the rule separately to each integration variable. The trapezoidal rule approximates the curve of a function with linear segments and computes the area under these segments. On the other hand, Simpson's rule approximates the function with parabolic segments, which can provide more accuracy if the integrand is well-behaved and smooth. However, it is important to note that while Simpson’s rule requires an even number of intervals, the trapezoidal rule does not have this restriction.
Although the analytical method is more accurate than both numerical methods, as it can provide exact results, numerical methods are used when the analytical approach is intractable or difficult to apply. They are also beneficial when dealing with integrands that are tabulated and when the integral cannot be expressed in terms of elementary functions, calling for a numerical approach.