The 3-point Gaussian integration formula is given by:
$$\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^3 A_i f(x_i)$$
with $x_1 = 0, x_2 = -x_3 = -\sqrt{3/5}$, and $A_1 = \frac{8}{9}, A_2 = A_3 = \frac{5}{9}$. This formula exactly integrates:
Options:
(A) $f(x) = 5 - x^7$
(B) $f(x) = 2 + 3x + 6x^4$
(C) $f(x) = 13 + 6x^3 + x^6$
(D) $f(x) = e^{-x^2}$
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.3)**
**1. Degree of Precision of Gauss-Legendre Quadrature**:
An $n$-point Gaussian quadrature rule (Gauss-Legendre integration) over the standard domain $[-1, 1]$ has a polynomial degree of precision equal to:
$$\text{Degree of Precision} = 2n - 1$$
For a $3$-point rule ($n = 3$):
$$\text{Degree of Precision} = 2(3) - 1 = 5$$
This means the formula evaluates any polynomial of degree up to $5$ with **zero error (exact integration)**.
**2. Option Evaluation**:
- **(A) $f(x) = 5 - x^7$**: Polynomial of degree 7 ($7 > 5$). $\implies$ Not exact.
- **(B) $f(x) = 2 + 3x + 6x^4$**: Polynomial of degree 4 ($4 \le 5$). $\implies$ **EXACT**.
- **(C) $f(x) = 13 + 6x^3 + x^6$**: Polynomial of degree 6 ($6 > 5$). $\implies$ Not exact.
- **(D) $f(x) = e^{-x^2}$**: Non-polynomial transcendental function. $\implies$ Not exact.
**3. Official Master Key**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: An $n$-point Gaussian quadrature rule integrates polynomials of degree up to $2n - 1$ exactly. For $n = 3$, all polynomials with degree $\le 5$ are exact.