Question

Simpson’s one-third rule will give the exact value of the integral \[ I=\int_a^b \left[ b_0 + b_1 x + b_2 x^2 + \cdots + b_n x^n \right] dx \] (where \(a, b, b_0, b_1, b_2, \ldots, b_n\) are numeric constants), if the values of \(n\) are:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Simpson’s 1/3 rule is exact for cubic and lower-degree polynomials; higher-degree polynomials require more advanced numerical methods.

Answer 1

The Correct Option is A, B, C

Simpson’s one-third rule is a numerical integration method that gives exact results for polynomials up to degree 3. This is because the method is based on approximating the function with a quadratic polynomial within each subinterval, but the overall composite rule integrates cubic polynomials exactly.
Step 1: Understand the rule.
Simpson’s 1/3 rule is exact for polynomials of degree: - 0 (constant),
- 1 (linear),
- 2 (quadratic),
- 3 (cubic).
Step 2: Apply to the given choices.
Values of \(n\) for which Simpson’s rule gives the exact integral:
- \(n = 1\) – exact,
- \(n = 2\) – exact,
- \(n = 3\) – exact,
- \(n = 4\) – NOT exact.
Step 3: Conclusion.
Thus, the exact answers are \(n = 1, 2, 3\).