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 ________________.
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 ________________.
The percentage error \[ e = 100 \times \frac{(I_e - I_s)}{I_e} \] is ________________.
Show Hint
Simpson’s rule gives exact results for any polynomial up to cubic degree. For quadratics, the error is always zero.
Updated On: Dec 1, 2025
- 2.5
- 3.5
- 1.2
- 0
Show Solution
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The Correct Option is D
Solution and Explanation
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
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