Question

While evaluating \(\int_{a}^{b} f(x) \, dx\) using Trapezoidal rule and Simpson's one-third rule, \([a,b]\) is divided into n number of subintervals so that both the methods can be applied for the same division, then a possible value of n is

• A. \(5\)
• B. \(12\)
• C. \(7\)
• D. Any non-prime number

Hint:

Simpson's 1/3 rule \(\rightarrow\) \(n\) must be a multiple of 2 (even).
Simpson's 3/8 rule \(\rightarrow\) \(n\) must be a multiple of 3.
Trapezoidal rule \(\rightarrow\) \(n\) can be any positive integer.

Answer 1

The Correct Option is B

Concept:
• The Trapezoidal rule can work with any integer number of subintervals \(n \ge 1\).
• Simpson's one-third rule strictly requires the number of subintervals \(n\) to be an even number.
• For both methods to be applicable simultaneously with the same division, \(n\) must satisfy both constraints, meaning \(n\) must be an even integer.

Step 1: Evaluate the conditions for each option

• Option A: \(n = 5\) is odd (Simpson's 1/3 cannot be applied).
• Option B: \(n = 12\) is even (both methods can be applied).
• Option C: \(n = 7\) is odd (Simpson's 1/3 cannot be applied).
• Option D: "Any non-prime number" includes odd composites like \(9\) or \(15\), which fail the even constraint.

Step 2: Select the valid option conforming to both requirements
The only value that guarantees an even number of intervals is \(12\).