---
title: "Gauss quadrature numerical integration"
subject: "Finite Element Method"
chapter: 7
key: "semester_8/finite_element_method:gauss-quadrature-integration"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/gauss-quadrature-integration
site: BCT Engineering
---

# Gauss quadrature numerical integration

Finite Element Method · Chapter 7: Higher order Elements

- Asked in: year not given, 5 marks
- Video: [Gauss Quadrature | Numerical Integration | Finite Element Analysis (FEA)](https://www.youtube.com/watch?v=0TV2SpMEzNk) — The Mechanical Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/gauss-quadrature-integration
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-7

## Question

Explain Gauss quadrature as used in finite element integration. Evaluate $I = \displaystyle\int_0^2 (x^4 + 3x)\,dx$ using 2-point and 3-point Gauss quadrature and compare with the exact value.

## Short answer

Gauss quadrature approximates ∫_{-1}^{1} f dξ ≈ Σ w_i f(ξ_i), integrating polynomials up to degree 2n–1 exactly. After mapping x=1+ξ, exact I=12.4. 2‑point gives 12.222 (−1.4% error); 3‑point gives 12.400 (exact).

## Answer

#### Gauss quadrature

In isoparametric elements, the stiffness integral $\int [B]^T[D][B]\det J\,d\xi\,d\eta$ is rational in $\xi$ and $\eta$, so it is evaluated numerically over $-1 \le \xi \le 1$:
$$
\int_{-1}^{1} f(\xi)\,d\xi \approx \sum_{i=1}^{n} w_i\,f(\xi_i)
$$
An $n$-point rule has sampling points $\xi_i$ and weights $w_i$ chosen to integrate exactly a polynomial of degree up to $2n - 1$.

| $n$ | Points $\xi_i$ | Weights $w_i$ | Exact up to degree |
|---|---|---|---|
| 1 | 0 | 2 | 1 |
| 2 | $\pm0.57735$ | 1, 1 | 3 |
| 3 | $\pm0.77460$, 0 | 5/9, 5/9, 8/9 | 5 |

For two dimensions, the rule is applied in both directions, $\sum\sum w_iw_jf(\xi_i, \eta_j)$ ($2\times2$ gives 4 points for a 4-node element).

#### Numerical example

Change of limits: $x = 1 + \xi$, so $dx = d\xi$ and $f(\xi) = (1 + \xi)^4 + 3(1 + \xi)$.

**Exact value:**
$$
I = \left[\frac{x^5}{5} + \frac{3x^2}{2}\right]_0^2 = \frac{32}{5} + 6 = 12.4
$$

**2-point rule:** $\xi = \pm0.57735$

| $\xi$ | $x = 1 + \xi$ | $f$ | $w$ |
|---|---|---|---|
| $-0.57735$ | 0.42265 | $0.03191 + 1.26795 = 1.29986$ | 1 |
| $+0.57735$ | 1.57735 | $6.19035 + 4.73205 = 10.92236$ | 1 |

$$
I_2 = 1.29986 + 10.92236 = 12.222
$$
The error is $-1.4\%$; the rule is not exact because $x^4$ is of degree 4 > 3.

**3-point rule:** $\xi = -0.77460,\ 0,\ +0.77460$

| $\xi$ | $x$ | $f$ | $w$ |
|---|---|---|---|
| $-0.77460$ | 0.22540 | 0.67879 | 5/9 |
| 0 | 1.00000 | 4.00000 | 8/9 |
| $+0.77460$ | 1.77460 | 15.24121 | 5/9 |

$$
I_3 = \tfrac{5}{9}(0.67879 + 15.24121) + \tfrac{8}{9}(4.0) = 8.8444 + 3.5556 = 12.400
$$
This equals the exact value, since the polynomial is of degree 4 ≤ 5.

**Answer:** 2-point: $I = 12.222$ (error 1.4 %); 3-point: $I = 12.400$ (exact).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Up to what polynomial degree is an n-point Gauss rule exact?
   A. n + 1
   B. 2n - 1
   C. n
   D. 2n
   Answer: B. Points and weights are chosen to integrate polynomials up to degree 2n - 1 exactly (2 points: 3, 3 points: 5). (Key point: Exactness)

2. Why is the 2-point result for the integral of (x^4 + 3x) from 0 to 2 not exact?
   A. The weights are not 1
   B. x^4 has degree 4, above 3, so it is not integrated exactly (I = 12.222)
   C. The limits are not -1 to 1
   D. 3x is non-linear
   Answer: B. The 2-point rule is exact only up to cubics, giving 12.222 (error -1.4%) against the exact 12.4. (Key point: Two-point rule)

3. What are the 3-point abscissae and weights, and the result?
   A. xi = +-1, 0; weights 1/3, 4/3, 1/3; I = 12.4
   B. xi = +-0.7746, 0; weights 5/9, 8/9, 5/9; I = 12.400 (exact)
   C. xi = +-0.57735; weights 1, 1; I = 12.222
   D. xi = 0; weight 2; I = 12.0
   Answer: B. The 3-point rule is exact up to degree 5, so it integrates the quartic exactly and gives 12.4. (Key point: Three-point rule)

4. How is the interval 0 to 2 mapped to -1 to 1?
   A. x = 1 - 2 xi
   B. x = 1 + xi, so dx = d xi
   C. x = xi/2
   D. x = 2 xi
   Answer: B. The midpoint is 1 and the half-length is 1, so x = 1 + xi and dx = d xi. (Key point: Change of limits)

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