AMU 2026 — Mathematics PYQ

1. Understanding Numerical Integration Rules

When evaluating a definite integral ${\int }_a^{b}f(x)dx$ numerically, the actual curve $y=f(x)$ is approximated by a simpler polynomial over small intervals (subintervals). Different rules use polynomials of different degrees:

• Trapezoidal Rule: Approximates the curve using a straight line (a first-degree polynomial, $n=1$).

• Simpson's One-Third Rule: Approximates the curve using a quadratic polynomial (a second-degree polynomial, $n=2$).

• Simpson's Three-Eighths Rule: Approximates the curve using a cubic polynomial (a third-degree polynomial, $n=3$).

2. Geometry of Simpson's $\frac{1}{3}$ Rule

Simpson's $\frac{1}{3}$ rule approximates the area under the curve by grouping the subintervals into pairs (requiring an even number of intervals). For each pair of consecutive intervals containing three distinct points, a unique quadratic interpolating polynomial of the form:

$$y=\alpha x^2+\beta x+\gamma$$

is fitted through the points. Geometrically, a second-degree polynomial equation $y=\alpha x^2+\beta x+\gamma$ represents a parabola.

Conclusion

Because Simpson's $\frac{1}{3}$ rule substitutes the original function $y=f(x)$ with a series of approximating quadratic curves across successive intervals, the curve is explicitly assumed to be a parabola.

This directly matches option (b).