Question

Consider the definite integral \[ \int_{1}^{2} (4x^2 + 2x + 6)\, dx. \] Let I_e be the exact value. Using Simpson’s rule with 10 equal subintervals, the estimate is I_s.
The percentage error \[ e = 100 \times \frac{(I_e - I_s)}{I_e} \] is ________________.

• A. 2.5
• B. 3.5
• C. 1.2
• D. 0

Hint:

Simpson’s rule gives exact results for any polynomial up to cubic degree. For quadratics, the error is always zero.

Answer 1

The Correct Option is D

The integrand is a quadratic polynomial: \[ f(x) = 4x^2 + 2x + 6. \] Simpson's rule gives exact results for all polynomials of degree ≤ 3. Since the function is degree 2, Simpson’s approximation has zero error. Compute exact integral: \[ I_e = \int_1^2 (4x^2 + 2x + 6)\, dx = \left(\frac{4x^3}{3} + x^2 + 6x\right)_1^2 = 14. \] Simpson’s rule gives exactly the same value: \[ I_s = 14. \] Thus percentage error: \[ e = 100 \times \frac{14 - 14}{14} = 0. \] Final Answer: 0