Trapezoidal rule

Definition

The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, providing a more accurate estimate of the total area compared to methods like the midpoint rule. This method is particularly useful for functions that are difficult or impossible to integrate analytically, allowing for numerical estimates with manageable computational effort.

5 Must Know Facts For Your Next Test

1. The trapezoidal rule approximates the integral by calculating the area of trapezoids formed by connecting points on the function's graph with straight lines.
2. The accuracy of the trapezoidal rule improves as the number of subdivisions increases, reducing the width of each trapezoid.
3. 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 dividing the interval [$$a$$, $$b$$].
4. Error in the trapezoidal rule is dependent on the second derivative of the function being integrated; if this derivative is small, accuracy is increased.
5. The trapezoidal rule is often used as a first step in more complex numerical methods, providing a good estimate that can be refined using techniques like Simpson's Rule.

Review Questions

• How does the trapezoidal rule improve upon simpler methods like the midpoint rule for numerical integration?
  • The trapezoidal rule improves upon methods like the midpoint rule by using trapezoids to estimate area instead of rectangles. While the midpoint rule calculates height at just one point per interval, the trapezoidal rule considers both endpoints of each interval, allowing it to better account for changes in function behavior. This leads to generally more accurate approximations, especially when the function has curvature, as it effectively captures more of the area under the curve.
• Discuss how increasing the number of subdivisions affects the accuracy of the trapezoidal rule.
  • Increasing the number of subdivisions reduces the width of each individual trapezoid, which typically enhances the accuracy of the trapezoidal rule. As subdivisions increase, each trapezoid better fits the underlying curve of the function being integrated. This reduction in error occurs because smaller intervals can capture more detail about changes in function values, ultimately leading to a closer approximation of the true integral value.
• Evaluate the strengths and weaknesses of using the trapezoidal rule in comparison to Simpson's Rule for numerical integration tasks.
  • The trapezoidal rule is straightforward and easy to implement, making it useful for quick approximations. However, it may not always provide sufficient accuracy for functions with significant curvature because it uses linear segments to estimate areas. In contrast, Simpson's Rule employs parabolic segments which can closely align with curved functions, usually yielding better accuracy. Yet, Simpson's Rule requires an even number of intervals and may involve more complex calculations, making it less practical in certain scenarios. Ultimately, choosing between them depends on desired accuracy and computational efficiency.

Related terms

Numerical Integration: A set of techniques used to calculate the integral of a function when an analytical solution is difficult or impossible to obtain.

Midpoint Rule: A numerical integration method that estimates the area under a curve by using rectangles whose heights are determined by the value of the function at the midpoint of each interval.

Simpson's Rule:

A numerical integration method that uses parabolic segments instead of straight lines or trapezoids to approximate the area under a curve, often yielding more accurate results than the trapezoidal rule.