Question

The value of the integral \[ \int_1^3 \left( x^2 - 2x \right) \, dx \] obtained by using Simpson’s 1/3 rule with 4 subintervals is equal to \( \frac{n}{3} \). The value of \( n \) is .............

Hint:

Simpson’s 1/3 rule provides an accurate numerical approximation for definite integrals, especially when the number of subintervals is even. Always ensure that \( n \) is even when applying this rule.

Answer 1

To solve this, we use Simpson's 1/3 rule for numerical integration. The formula for Simpson’s 1/3 rule with \( n \) subintervals is given by: \[ I \approx \frac{b - a}{3n} \left[ f(a) + 4 \sum_{i=1}^{n-1} f(x_i) + f(b) \right], \] where \( a \) and \( b \) are the limits of integration, and \( f(x_i) \) represents the function evaluated at points between \( a \) and \( b \). For this question, we are asked to apply Simpson’s 1/3 rule with 4 subintervals, meaning \( n = 4 \). 
However, Simpson’s rule requires that the number of subintervals be even, so \( n \) in this case must be 2, as the total number of subintervals for Simpson's 1/3 rule is always halved. 
Therefore, the correct answer is \( n = 2 \).