GATE 2025 ME – Question 31
The values of a function $f$ obtained for different values of $x$ are shown in the table below.
| $x$ | 0 | 0.25 | 0.5 | 0.75 | 1.0 |
|---|---|---|---|---|---|
| $f(x)$ | 0.9 | 2.0 | 1.5 | 1.8 | 0.4 |
Using Simpson's one-third rule,
$$\int_0^1 f(x)\,dx \approx \_\_\_\_\_\_ \text{ (rounded off to 2 decimal places).}$$
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Show answer and explanation
Correct answer: 1.62 to 1.64
Explanation
With $h = 0.25$, Simpson's one-third rule gives $\frac{h}{3}\left[f_0 + 4f_1 + 2f_2 + 4f_3 + f_4\right] = \frac{0.25}{3}\left[0.9 + 8.0 + 3.0 + 7.2 + 0.4\right] = \frac{0.25}{3} \times 19.5 = 1.625$, which rounds to 1.63.