Part 18: General Quadrature Formula for UPSC ISS: The Mother of All Integration Rules

Welcome back to Part 18 of Module 5, updated for the UPSC ISS 2026 to 2027 cycle. Part 17 taught us rates of change from tables. Today we learn accumulated areas from tables: Numerical Integration, and it all begins with one master equation, the General Quadrature Formula, also called the Newton Cotes Quadrature Formula.

Numerical quadrature evaluates a definite integral $\int_a^b f(x)\,dx$ from discrete values of the integrand. Integrating the Newton Gregory Forward formula over $n$n equal parts gives the mother formula, and substituting $n = 1, 2, 3, 4, 6$ produces the Trapezoidal Rule, Simpson’s 1/3, Simpson’s 3/8, Boole’s Rule, and Weddle’s Rule respectively.

Exam context: 80 questions, 200 marks, 2 hours, 2.5 marks per question, about 0.83 deducted per wrong answer, and integration questions form the heart of the Numerical Analysis block. Several are pure theory on this very page.

This post is Part 18 of the Numerical Calculus Guide (Module 5), inside our UPSC ISS Numerical Analysis Complete Guide. Roadmap at the UPSC ISS hub.

The Story of Karthik and the Highway Land Survey

Karthik, an ISS officer in the Ministry of Road Transport and Highways, must report the exact area of an irregular tract of land bordered by a curving river, so farmers can be compensated fairly for the new expressway.

A textbook problem would hand him the river’s equation to integrate. The real river obeys no equation. His surveyors instead give him the land’s width measured at equal 10 metre intervals. Karthik fits a polynomial through the discrete widths using Newton’s Forward formula, integrates the polynomial, and reports the area within minutes. That is numerical quadrature at work.

Quadrature and Cubature

Numerical quadrature is the evaluation of a definite integral of a function of a single variable from discrete values, geometrically an area. When the object is a double integral in two variables, the process is called mechanical cubature, geometrically a volume. UPSC has framed direct definitional questions on the distinction.

The Mother Formula

Divide the range $(a, b)$ into $n$ equal parts of width $h$, giving points $x_0, x_0 + h, \dots, x_0 + nh$. Approximating $y$ by Newton’s Forward formula and integrating from $x_0$​ to $x_n$​:$$I = h\left[n\,y_0 + \frac{n^2}{2}\,\Delta y_0 + \left(\frac{n^3}{3} – \frac{n^2}{2}\right)\frac{\Delta^2 y_0}{2!} + \left(\frac{n^4}{4} – n^3 + n^2\right)\frac{\Delta^3 y_0}{3!} + \dots\right]$$

The family tree. Every famous rule is a child of this equation:

$n = 1$ gives the Trapezoidal Rule. $n = 2$ gives Simpson’s 1/3 Rule. $n = 3$ gives Simpson’s 3/8 Rule. $n = 4$ gives Boole’s Rule. $n = 6$ gives Weddle’s Rule.

Cote’s Numbers: The Hidden Goldmine

The Newton Cotes formula can also be written by weighting the entries directly: $I = nh\sum_{k=0}^{n} y_k\, C_k^n$​, where the weights $C_k^n$ are the Cote numbers, depending only on $n$ and $k$ and never on the function.

Two must memorize properties.

1. Symmetry: $C_k^n = C_{n-k}^n$​. The coefficients mirror from both ends, so you only ever compute half of them.
2. Unit sum: $\sum_{k=0}^{n} C_k^n = 1$ for every $n$, without exception.

StatChakravyuh Pro Tips

1. The one second Cote hack. If a question asks for the sum of Cote coefficients for an 8th degree Newton Cotes formula, or any degree at all, the answer is 1. Tick and move on.
2. The root formula fact. Asked which interpolation formula the general quadrature is derived from, the answer is strictly the Newton Gregory Forward formula.
3. Equidistant only. The Newton Cotes family assumes strictly equal intervals. Unequal spacing requires integrating Lagrange’s polynomial instead.

Solved Derivation Masterclass

Question: Derive $\int_{x_0}^{x_1} y\,dx$ by putting $n = 1$ in the General Quadrature Formula.

Solution.

1. With $n = 1$, the fitted curve through 2 points is a straight line, so second and higher differences vanish.
2. Substitute: $I = h\left[1 \cdot y_0 + \dfrac{1}{2}\Delta y_0\right]$.
3. Use $\Delta y_0 = y_1 – y_0$​: $I = h\left[y_0 + \dfrac{1}{2}(y_1 – y_0)\right] = h \cdot \dfrac{2y_0 + y_1 – y_0}{2}$​​.

$$I = \frac{h}{2}(y_0 + y_1)$$

Result: the exact Trapezoidal Rule, produced by pure algebra with no calculus. Part 19 explores this child in full depth.

Common Traps to Avoid

Trap 1: Attaching Weddle’s Rule to $n = 5$. Weddle comes from $n = 6$; the number 5 is the planted distractor.

Trap 2: Believing Cote numbers depend on the function being integrated. They depend only on $n$ and $k$.

Trap 3: Applying Newton Cotes to unequally spaced data. The derivation itself assumes a constant $h$.

Frequently Asked Questions

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Coming up in Part 19: the first child of the family, the Trapezoidal Rule, and the local versus global error distinction that UPSC loves to test.