The GATE Grind

GATE 2020 CE (CE2) – Question 13

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 1 mark · Multiple choice

Estimate $\int_0^1(5x^3+4x^2+3x+2)\,dx$ by rectangular, trapezoidal and Simpson’s rules with a common step size. Which is TRUE?

  1. Simpson’s and rectangular rules both give nonzero error.
  2. Simpson’s, rectangular and trapezoidal rules all give nonzero error.
  3. Only rectangular rule gives zero error.
  4. Only Simpson’s rule gives zero error.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) Only Simpson’s rule gives zero error.

Explanation

Simpson’s rule integrates polynomials of degree at most3 exactly,so its error for this cubic is zero. Rectangle and trapezoid rules are not exact here;the integrand is increasing and strictly convex. Thus **D**. The usual valid Simpson subdivision pattern is understood.