Question

While applying Simpson's three eighth rule for numerical integration, the number of sub-intervals should be taken as multiple of

• A. 2
• B. 3
• C. 8
• D. 6

Hint:

Remember the rule names as clues: Simpson's 1/2-like rule (1/3rd rule) requires multiples of 2, whereas Simpson's 3/8th rule contains a 3 in its name and requires multiples of 3.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Numerical integration methods approximate the area under a curve by dividing the integration interval into a specific number of equal sub-intervals.

Step 2: Detailed Explanation:
Different Newton-Cotes formulas require different constraints on the number of sub-intervals ($n$):
1. Trapezoidal Rule: Based on linear interpolation, with no restrictions on the number of sub-intervals ($n$ can be any integer).
2. Simpson's 1/3rd Rule: Based on approximating the function with a second-degree parabola over pairs of intervals. This requires $n$ to be an even number (a multiple of 2).
3. Simpson's 3/8th Rule: Based on fitting a third-degree cubic polynomial over groups of three intervals. To apply this rule across the entire interval, the total number of sub-intervals ($n$) must be a multiple of 3.
If these constraints are not met, the respective numerical integration rule cannot be applied across the entire range.

Step 3: Final Answer:
The number of sub-intervals for Simpson's 3/8th rule must be a multiple of 3.