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GATE 2022 ME (ME2) – Question 13

Engineering Mathematics · Numerical Methods: Algebraic equations, integration by trapezoidal and Simpson rules, ODE methods · 1 mark · Multiple choice

Consider the definite integral $\int_1^2(4x^2+2x+6)\,dx$. Let $I_e$ be the exact value of the integral. If the same integral is estimated using Simpson’s rule with 10 equal subintervals, the value is $I_s$. The percentage error is defined as $e=100\times(I_e-I_s)/I_e$. The value of $e$ is

  1. 2.5
  2. 3.5
  3. 1.2
  4. 0

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Correct answer: (D) 0

Explanation

**Key fact.** Composite Simpson's 1/3 rule has an error term proportional to the **fourth derivative** of the integrand. It therefore integrates every polynomial of degree **3 or less exactly**.

The integrand $4x^2+2x+6$ has degree 2, so its fourth derivative is zero and the rule gives the exact value, $I_s=I_e$ (for any number of subintervals, here 10).

$$e=100\times\frac{I_e-I_s}{I_e}=\mathbf{0}.$$