Understanding the Trapezoidal Rule
The Trapezoidal Rule is a numerical integration technique that approximates the area under a curve by dividing it into trapezoids. Its effectiveness depends largely on the order of the polynomial being integrated.
Integration of Polynomials
- Trapezoidal Rule is particularly effective for linear (1st order) polynomials, as it assumes a straight-line approximation between two points.
Performance on Higher Order Polynomials
- 1st Order (Linear): The trapezoidal rule perfectly integrates linear functions because the area under a straight line is precisely captured by the trapezoid formed between two points.
- 2nd Order (Quadratic): While the trapezoidal rule can still provide reasonable approximations, it introduces some error due to the curvature of the parabola.
- 3rd Order (Cubic): The error increases significantly for cubic functions as the trapezoidal rule cannot adequately capture the curvature, leading to substantial inaccuracies.
- 4th Order (Quartic): Integration accuracy further deteriorates with quartic polynomials, as the trapezoidal rule fails to account for the increased complexity of the function's shape.
Conclusion
The trapezoidal rule is best suited for integrating 1st order polynomials because:
- It perfectly matches the linear function's area under the curve.
- As the order increases, the approximation errors grow due to the method's inherent limitations in capturing non-linear shapes.
Thus, the correct choice is option D (1st order), as it allows for the most accurate integration using the Trapezoidal Rule.