Will I ever need the actual Simpson formula in the exam?

Yes, whenever the integrand is not a polynomial, like sin⁡x\sin x or exe^x, or the degree exceeds 3, like x4x^4. In such questions the values are usually kept simple for manual work.

What is the exactness degree of Weddle's Rule?

Weddle's Rule is exact for polynomials of degree 5 or lower. A statement claiming exactness for a 4th degree polynomial is therefore correct.

How do I predict whether an integral will carry an error?

Compare the degree of the integrand with the rule's exactness ceiling. Trapezoidal on a quadratic has an error; Simpson's 1/3 on a 4th degree polynomial has an error; Weddle on a 4th degree polynomial has none.

Quadrature PYQ Exactness: Solve UPSC ISS Integration in 10 Seconds

Welcome to Part 22, the finale of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. Across the last four parts we built the entire integration toolkit, from straight lines to parabolas to Weddle’s precision. Knowing the formulas is half the battle. The other half: knowing when you never need them.

A quadrature rule is exact for a polynomial when its truncation error vanishes, making the numerical answer identical to the analytical integral. With calculators banned, a manual 10 interval Simpson computation costs 10 minutes and invites a fractional slip worth minus 0.83 marks. An analysis of past papers shows the examiners repeatedly design questions to reward the exactness properties instead. Today is a rapid fire session on exactly those questions.

This post is Part 22 of the Numerical Calculus Guide (Module 5), inside our UPSC ISS Numerical Analysis Complete Guide. Roadmap at the UPSC ISS hub.

Table of Contents

The Exactness Cheat Code Summary

Memorize this table permanently. It is the single highest value block of Module 5.

Trapezoidal Rule: exact only for linear polynomials, degree 1. Simpson’s 1/3 Rule: exact for quadratic and cubic polynomials, up to degree 3. Simpson’s 3/8 Rule: exact for quadratic and cubic polynomials, up to degree 3. Weddle’s Rule: exact for polynomials of degree 5 or lower, so a 4th degree polynomial is handled perfectly.

The universal decision procedure: read the degree of the integrand, compare it with the rule’s exactness ceiling. Degree within the ceiling means zero error and the numerical answer equals the plain analytical integral; degree above the ceiling, or a non polynomial integrand, means the formula must actually be used.

Which rule’s exactness degree do you find hardest to retain? Tell us and we will share a memory hook.

Which rule's exactness degree do you find hardest to retain? Tell us and we will share a memory hook.x

Case Study 1: The Error Difference Trap

The pattern: let $a$ a be the exact value $\int_0^{10} x^2\,dx = \left[\dfrac{x^3}{3}\right]_0^{10} = \dfrac{1000}{3}$ , and let $b$ be the value computed by Simpson’s 1/3 rule with 10 equal subdivisions. Find $|a – b|$ .

The trap: computing all 11 ordinates and grinding the composite formula, several minutes of fraction work under pressure.

The 5 second logic: the integrand $x^2$ has degree 2, inside Simpson’s degree 3 ceiling, so $b$ equals $a$ exactly and $|a – b| = 0$ . Marks secured without touching a rough sheet.

Case Study 2: The Direct Integration Bypass

PYQ (UPSC ISS 2020 cycle, as recorded in our PYQ archive): $\int_0^1 (3x^2 + 25)\,dx$ is evaluated by Simpson’s one third rule with $h = \tfrac{1}{4}$ . The value obtained is:

(a) 23 (b) 24.56 (c) 25.33 (d) 26

The 10 second logic: the $h = \tfrac{1}{4}$ invites a fractional table at $x = 0, 0.25, 0.5, 0.75, 1$ . Ignore it. The integrand has degree 2, inside the ceiling, so the Simpson value equals the analytical value: $\left[x^3 + 25x\right]_0^1 = 1 + 25 = 26$ .

Final Answer: (d) 26.

Case Study 3: The Statement Based Theory Trap

PYQ (UPSC ISS 2025 cycle, as recorded in our PYQ archive): Consider the statements:

I. The Trapezoidal rule gives an exact result for a linear polynomial. II. The Trapezoidal rule gives an exact result for a quadratic polynomial. III. Simpson’s one third rule gives an exact result for a cubic polynomial. IV. Simpson’s one third rule gives an exact result for a quadratic polynomial.

Which statements are correct?

The logic: the Trapezoidal rule draws straight lines, so its error vanishes only for degree 1; statement I is true and statement II is false. Simpson’s 1/3 carries the degree 3 exactness through its fourth derivative error term, so statements III and IV are both true.

Final Answer: I, III, and IV.

When the Trick Does Not Apply

Honesty matters as much as speed. The exactness bypass fails in two situations: the integrand is not a polynomial at all, such as $\sin x$ , $e^x$ , or $\log x$ ; or the polynomial degree exceeds the rule’s ceiling, such as $x^4$ under Simpson’s 1/3. In those cases the formula must genuinely be applied, and UPSC then keeps the arithmetic gentle, typically $h = 1$ with small integer ordinates. Recognizing which regime a question belongs to is itself the skill this part trains.

Frequently Asked Questions

Will I ever need the actual Simpson formula in the exam?

Yes, whenever the integrand is not a polynomial, like $\sin x$ or $e^x$ , or the degree exceeds 3, like $x^4$ . In such questions the values are usually kept simple for manual work.
What is the exactness degree of Weddle’s Rule?

Weddle’s Rule is exact for polynomials of degree 5 or lower. A statement claiming exactness for a 4th degree polynomial is therefore correct.
How do I predict whether an integral will carry an error?

Compare the degree of the integrand with the rule’s exactness ceiling. Trapezoidal on a quadratic has an error; Simpson’s 1/3 on a 4th degree polynomial has an error; Weddle on a 4th degree polynomial has none.

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Module 5, the biggest scoring module, is complete. Next stop, Module 6: Difference Equations and the Summation of Series, the quick win module of the syllabus.

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Part 22: Quadrature PYQs in 10 Seconds: The Exactness Cheat Codes for UPSC ISS