Trapezoidal rule

Definition

The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. By dividing the area under the curve into trapezoids rather than rectangles, this method provides a more accurate estimate of the integral by considering the linear interpolation between function values at specified points. This technique is particularly useful when dealing with complex functions that are difficult to integrate analytically.

5 Must Know Facts For Your Next Test

1. The trapezoidal rule improves accuracy by averaging the values of the function at the endpoints of each subinterval, leading to better estimates compared to simple rectangular methods.
2. The formula for the trapezoidal rule is given by $$T = \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right)$$, where 'n' is the number of subdivisions, and 'x_i' are the points at which the function is evaluated.
3. The error in the trapezoidal rule decreases quadratically with an increase in the number of subintervals, meaning that doubling the number of intervals will roughly reduce the error by a factor of four.
4. This rule can be applied iteratively, refining estimates by increasing subdivisions until a desired accuracy is achieved.
5. While useful, the trapezoidal rule can struggle with functions that exhibit high curvature or discontinuities, leading to larger errors compared to other methods like Simpson's Rule.

Review Questions

• How does the trapezoidal rule compare to other numerical integration techniques in terms of accuracy and application?
  • The trapezoidal rule tends to be more accurate than simpler methods like the rectangular rule because it accounts for the linear change in function values across subintervals. However, it may not perform as well as Simpson's Rule when dealing with functions that have significant curvature. The choice between these methods often depends on the nature of the function being integrated and the level of precision required.
• Discuss how you would apply the trapezoidal rule in practice and what considerations you should keep in mind regarding its limitations.
  • To apply the trapezoidal rule, you would start by defining your function and determining the interval over which to integrate. You then decide how many subdivisions to use, evaluating the function at each endpoint and summing their contributions according to the trapezoidal formula. It’s important to consider that while this method is straightforward, it may yield less accurate results for functions with steep slopes or discontinuities, necessitating more subdivisions for better accuracy.
• Evaluate how increasing the number of subdivisions affects both accuracy and computational efficiency when using the trapezoidal rule for numerical integration.
  • Increasing the number of subdivisions generally improves accuracy in the trapezoidal rule due to a reduction in approximation error, as smaller intervals provide a closer representation of the area under the curve. However, this improvement comes at a cost to computational efficiency since more evaluations of the function are needed. Finding a balance between desired accuracy and acceptable computation time is crucial, especially when integrating complex functions.

Related terms

Numerical Integration: A mathematical process used to calculate the approximate value of an integral using numerical methods rather than analytical approaches.

Simpson's Rule: Another numerical integration technique that approximates the integral of a function by using parabolic segments instead of linear segments.

Definite Integral: An integral that calculates the area under a curve between two specified limits, providing a numerical value that represents this area.