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GATE 2023 CH – Question 37

Engineering Mathematics · Numerical Methods: Algebraic equations, integration, ODEs and finite-difference methods · 2 marks · Multiple choice

Simpson's one-third rule is used to estimate the definite integral $I = \int_{-1}^{1}\sqrt{1 - x^2}\,dx$ with an interval length of 0.5. Which one of the following is the CORRECT estimate of $I$ obtained using this rule?

  1. $\frac{1}{3} - \frac{1}{\sqrt{3}}$
  2. $\frac{1}{3} + \frac{2}{\sqrt{3}}$
  3. $\frac{1}{3} + \frac{1}{\sqrt{3}}$
  4. $\frac{1}{3} - \frac{2}{\sqrt{3}}$

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Correct answer: (B) $\frac{1}{3} + \frac{2}{\sqrt{3}}$

Explanation

The points are $x = -1, -0.5, 0, 0.5, 1$ with $f = 0, \frac{\sqrt{3}}{2}, 1, \frac{\sqrt{3}}{2}, 0$. Simpson's rule: $\frac{h}{3}\left[f_0 + 4f_1 + 2f_2 + 4f_3 + f_4\right] = \frac{0.5}{3}\left[0 + 2\sqrt{3} + 2 + 2\sqrt{3} + 0\right] = \frac{1}{6}(2 + 4\sqrt{3}) = \frac{1}{3} + \frac{2}{\sqrt{3}}$ (about 1.488).