What is the mandatory condition for Simpson's 1/3 rule?

The range must be divided into an even number of equal sub intervals, which means the number of given ordinates must be odd, such as 3, 5, or 7 points.

How does the geometry differ from the Trapezoidal rule?

The Trapezoidal rule assumes a straight line between two points. Simpson's 1/3 assumes a parabola, a degree 2 polynomial, through every three consecutive points.

Why is a degree 2 rule exact for degree 3 polynomials?

The error term contains the fourth derivative, and the fourth derivative of any cubic is strictly zero, so the error vanishes for both quadratics and cubics.

What is the order of the error?

The global error is O(h4)O(h^4) with expression −(b−a)h4180f(4)(ξ)-\dfrac{(b-a)h^4}{180}f^{(4)}(\xi); the local error over one pair of intervals is O(h5)O(h^5).

Can I use the rule with 6 sub intervals?

Yes. Six is even, so it qualifies, and you need the 7 ordinates y0y_0 through y6y_6.

Simpson 1/3 Rule UPSC ISS: Formula, Even Rule, Degree 3 Trick

Welcome back to Part 20 of Module 5, the single most important part of this module, updated for the UPSC ISS 2026 to 2027 cycle. The Trapezoidal Rule draws straight lines and pays for it with an $O(h^2)$ error. The upgrade: replace straight lines with parabolas. That upgrade is Simpson’s 1/3 Rule, and hidden inside it lives the most valuable shortcut of the entire unit, the Degree 3 Exactness trick.

The formula: $I = \dfrac{h}{3}\left[(y_0 + y_n) + 4(\text{odd ordinates}) + 2(\text{even middle ordinates})\right]$ , valid strictly for an even number of sub intervals. Its global error is of order $O(h^4)$ and depends on the fourth derivative, which is why the rule is perfectly exact for every polynomial up to degree 3.

The stakes: 80 questions, 200 marks, 2 hours, 2.5 marks each, about 0.83 deducted per wrong answer. Both solved PYQs below are from the 2020 cycle and reward the trick with full marks in seconds.

This post is Part 20 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 Story of Sameer and the Highway Arch

Sameer, an ISS officer on a national highway project, must estimate the concrete needed for a parabolic arched bridge. The engineers hand him the arch heights at equal intervals.

The Trapezoidal Rule would connect the heights with straight chords, chopping the top off the curve and underestimating the area. Simpson’s 1/3 Rule instead fits a genuine parabola through every three consecutive points. For an arch that actually is a parabola, the fit is not an approximation at all; it wraps the shape exactly, and Sameer’s area comes out with zero error.

Have you ever estimated a curved shape with straight lines and landed far from the truth? Share the story in the comments.

Have you ever estimated a curved shape with straight lines and landed far from the truth? Share the story in the comments.x

The Core Formula

Setting $n = 2$ in the General Quadrature Formula and neglecting third and higher differences gives, for an even number of sub intervals of width $h$ : $I = \frac{h}{3}\left[(y_0 + y_n) + 4(y_1 + y_3 + y_5 + \dots) + 2(y_2 + y_4 + y_6 + \dots)\right]$

In simple words: a third of the gap, times the sum of the first and last entries, plus four times the sum of the odd position ordinates, plus twice the sum of the even position middle ordinates. The coefficient pattern reads $1, 4, 2, 4, 2, \dots, 4, 1$ .

The Ultimate Trap: The Even Constraint

The rule fits a parabola across 3 points, which covers 2 intervals, so the data must split into pairs of intervals. The number of sub intervals nn n must strictly be even. The mirror statement, equally tested: the number of ordinates, that is data points, must then be odd. Confusing intervals with points is the oldest trap in this topic. Six intervals are fine and need seven points.

Error Bound and the Degree 3 Exactness Trick

The global error: $E_{\text{global}} = -\frac{(b – a)\,h^4}{180}\,f^{(4)}(\xi)$

Two facts jump out. The order is $O(h^4)$ , far better than the Trapezoidal $O(h^2)$ , with a local error of $O(h^5)$ per pair of strips. And the error depends entirely on the fourth derivative.

The cheat code. Take any cubic polynomial, like $y = x^3 + 2x$ . Its fourth derivative is exactly zero. So the error term vanishes completely, and a rule derived from a degree 2 parabola delivers a 100 percent exact result for every polynomial up to degree 3.

A degree 2 construction that is secretly exact for degree 3. Drop a mind blown if this surprised you.

A degree 2 construction that is secretly exact for degree 3. Drop a mind blown if this surprised you.x

Solved PYQ Masterclass

PYQ 1, the integration bypass (UPSC ISS 2020 cycle, as recorded in our PYQ archive): The integral $\int_0^1 (3x^2 + 25)\,dx$ 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 $h = \tfrac{1}{4}$ is bait, inviting you to tabulate five fractional ordinates. Instead, read the integrand: $3x^2 + 25$ is degree 2, and Simpson’s 1/3 is exact up to degree 3, so the numerical value equals the analytical integral. Basic calculus: $\left[x^3 + 25x\right]_0^1 = 1 + 25 = 26$ .

Final Answer: (d) 26.

PYQ 2, the error difference (UPSC ISS 2020 cycle, as recorded in our PYQ archive): Let $A = \int_0^{10} x^2\,dx$ , and let $B$ be its value computed by Simpson’s 1/3 rule with 10 equal subdivisions. What is $|A – B|$ ?

The 5 second logic. $|A – B|$ is simply the error. The integrand $x^2$ has degree 2, within the exactness limit of 3, so $B$ equals $A$ exactly.

Final Answer: $|A – B| = 0$ . Ten subdivisions of arithmetic avoided entirely.

Common Traps to Avoid

Trap 1: Counting data points as intervals. Even intervals means odd ordinates; a table of 7 points has 6 intervals and qualifies.

Trap 2: Extending the exactness to degree 4. The fourth derivative of $x^4$ is a non zero constant, so $x^4$ carries a genuine error.

Trap 3: Building the fractional table when the integrand is a polynomial of degree 3 or below. The exactness trick makes the table pointless.

Frequently Asked Questions

What is the mandatory condition for Simpson’s 1/3 rule?

The range must be divided into an even number of equal sub intervals, which means the number of given ordinates must be odd, such as 3, 5, or 7 points.
How does the geometry differ from the Trapezoidal rule?

The Trapezoidal rule assumes a straight line between two points. Simpson’s 1/3 assumes a parabola, a degree 2 polynomial, through every three consecutive points.
Why is a degree 2 rule exact for degree 3 polynomials?

The error term contains the fourth derivative, and the fourth derivative of any cubic is strictly zero, so the error vanishes for both quadratics and cubics.
What is the order of the error?

The global error is $O(h^4)$ with expression $-\dfrac{(b-a)h^4}{180}f^{(4)}(\xi)$ ; the local error over one pair of intervals is $O(h^5)$ .
Can I use the rule with 6 sub intervals?

Yes. Six is even, so it qualifies, and you need the 7 ordinates $y_0$ through $y_6$ .

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Coming up in Part 21: intervals in multiples of 3 and 6, with Simpson’s 3/8 Rule and the precision champion, Weddle’s Rule.

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Part 20: Simpson’s 1/3 Rule for UPSC ISS: The Degree 3 Exactness Trick