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GATE 2023 CE (CE1) – Question 36

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 2 marks · Multiple choice

A function $f(x)$, that is smooth and convex-shaped between interval $(x_l, x_u)$ is shown in the figure. This function is observed at odd number of regularly spaced points. If the area under the function is computed numerically, then _____.

an arch-shaped curve of $f(x)$ above the axis between $x_l$ and $x_u$, with the approximating polyline (trapezoids) drawn below the curve at 9 regularly spaced points.
  1. the numerical value of the area obtained using the trapezoidal rule will be less than the actual
  2. the numerical value of the area obtained using the trapezoidal rule will be more than the actual
  3. the numerical value of the area obtained using the trapezoidal rule will be exactly equal to the actual
  4. with the given details, the numerical value of area cannot be obtained using trapezoidal rule

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Show answer and explanation

Correct answer: (A) the numerical value of the area obtained using the trapezoidal rule will be less than the actual

Explanation

As drawn, the curve is an arch, bulging above its chords. The trapezoidal rule replaces the curve by straight chords, which lie below the curve, so it underestimates the area. (An odd number of points means an even number of intervals, so the rule can be applied.)