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Numerical integration is a way to calculate the area under a curve when we either do not have an analytical formula or it is difficult to integrate. It helps us approximate the integral of a function that might otherwise be quite tricky to solve. Think of it as a strategy to find the total sum of values represented by a continuous function over a specific interval.

In this context, we use a variety of methods to achieve this approximation. One popular method is Simpson's Rule, which can accurately approximate the integral of polynomial functions or those that resemble them closely. Using Simpson's Rule, we apply known integrals of simple, easy-to-handle functions to approximate more complex integrals.

When applying numerical integration to the problem of a rocket's vertical distance, each velocity function over its time intervals is evaluated. We split these intervals into smaller sections (subintervals) and apply our numerical integration formula to find the distance traveled over each part.

In calculus and numerical analysis, approximation techniques are used to get a close estimate of a function's integral, which can oftentimes be difficult or impossible to determine directly. Simpson's Rule is a widely used approximation technique that finds the integral using parabolic arcs, rather than straight lines as seen in other methods.

Simpson's Rule assumes that within a small interval, a quadratic or second-degree polynomial can approximate the function. This is because parabolas provide a better fit for curved graphs than straight lines do, making this technique more accurate, particularly when dealing with polynomials of degree three or lower. This technique is especially useful across multiple subintervals, as seen in the step-by-step solution for estimating the distance travelled by the rocket.

To use this technique efficiently, each interval is divided into an even number of subintervals. The rule then provides a formula where you sum up values derived from the function being integrated, adjusted by coefficients (including 1, 4, and 2) specific to Simpson's Rule. The final result offers a good approximation of the actual integral over the desired range.

In the realm of numerical methods, subintervals play a crucial role in breaking down a complex range into smaller, more manageable parts. This division helps to increase accuracy when approximating the integral of a function over a certain interval. Simpson’s Rule, like many numerical methods, uses this technique to achieve better accuracy in its approximations.

In the exercise given, the time intervals are from 0 to 10, 10 to 20, and 20 to 30. For each of these, we further break them down into four subintervals. These subdivisions are crucial because they help us apply Simpson's Rule more effectively. The smaller our subintervals, the closer our approximation is likely to be to the real integral value.

Breaking intervals into equal subintervals and applying consistent coefficients helps keep calculations straightforward, and results more precise. When using Simpson's Rule, consistent application across all subintervals ensures an accurate estimate of the integral across the entire original range.