When we look at the Trapezoidal Rule and Simpson's Rule for finding areas under curves, we see some big differences in how well they work. Let’s break it down:

1. Trapezoidal Rule:

• This method finds the area under a curve by splitting it into trapezoids.
• It’s pretty easy to understand, but it can have trouble with curves that bend a lot.
• If you use more sections, it gets better, but sometimes it can either guess too low or too high, depending on how the curve acts.

2. Simpson's Rule:

• On the other hand, Simpson's Rule usually gives a more accurate answer. It uses curved shapes called parabolas instead of straight lines.
• This rule needs to work with an even number of sections, which helps it follow the curve properly.
• It’s especially good for smooth curves, giving excellent results.

Conclusion:
From what I’ve seen, both methods can be useful. However, Simpson's Rule often gives a better answer, especially with complicated curves. When you're working on AP Calculus problems, getting the right answer really matters!