When we're working with numerical integration, picking the right size for our intervals is really important. This choice affects how accurately we can estimate the area under a curve, which is what we are trying to do when we integrate a function.

Imagine we have a function that changes quickly. If we use a method like the Trapezoidal Rule or Simpson's Rule with a big interval, we might overlook important details that happen between the starting and ending points. This can cause big mistakes in our results. On the other hand, using smaller intervals helps us capture these changes better, leading to more accurate results. But remember, it’s not just about using smaller intervals; it also matters how these intervals work with the method we are using.

Here’s how the size of the intervals affects accuracy:

1. Trapezoidal Rule:

   • This method connects points with straight lines to guess the area.
   • If our intervals are large, the straight lines might not represent curved parts of the function well.
   • Smaller intervals create more sections, which helps to reflect the true shape of the function better.

2. Simpson's Rule:

   • This method uses curved shapes called parabolas to fit the function.
   • It's most accurate when the function is smooth and changes gradually.
   • If the intervals are too big, we end up with fewer parabolic sections, which can lead to more error.

Although smaller intervals help improve accuracy, they also require more calculations. So, we need to find a good balance between:

• What kind of function we are working with.
• How much error is acceptable.
• The computing power we have available.

In conclusion, while smaller intervals can lead to better accuracy, we must also think about the cost and time it takes to calculate. The best interval size changes from one problem to another, so we need to have a smart plan when using numerical integration methods.