Using Trapezoidal Approximations

Introduction

Key Concepts

1. Understanding Definite Integrals

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. Mathematically, it is expressed as:

where \(f(x)\) is the function being integrated, and \([a, b]\) is the interval of integration. The definite integral calculates the net area between the curve \(f(x)\) and the \(x\)-axis from \(x = a\) to \(x = b\).

2. Riemann Sums

Riemann sums are techniques for approximating the value of a definite integral. They partition the interval \([a, b]\) into smaller subintervals and sum the areas of rectangles or other shapes that approximate the region under the curve.

• Left Riemann Sum: Uses the left endpoint of each subinterval to determine the height of rectangles.
• Right Riemann Sum: Uses the right endpoint of each subinterval for rectangle heights.
• Midpoint Riemann Sum: Utilizes the midpoint of each subinterval for determining rectangle heights.

While Riemann sums provide a foundation for numerical integration, they can be inefficient and less accurate, especially with fewer subintervals.

3. The Trapezoidal Rule

The trapezoidal rule enhances the basic Riemann sum by approximating the area under the curve \([a, b]\) using trapezoids instead of rectangles. This method increases accuracy by considering the slope of the function over each subinterval.

The formula for the trapezoidal rule is:

where:

• \(n\) = number of subintervals
• \(\Delta x = \frac{b - a}{n}\) = width of each subinterval
• \(x_0, x_1, \ldots, x_n\) = endpoints of subintervals

This formula calculates the area by summing the areas of trapezoids formed between each pair of consecutive points on the curve.

4. Derivation of the Trapezoidal Rule

The trapezoidal rule can be derived by approximating the function \(f(x)\) with a linear function over each subinterval. By integrating the linear approximation, we obtain the area of a trapezoid, which serves as an approximation for the area under \(f(x)\).

The accuracy of the trapezoidal rule depends on the number of subintervals \(n\). Increasing \(n\) generally leads to a more accurate approximation as the trapezoids better conform to the curve.

5. Error Estimation

The error in the trapezoidal approximation can be estimated using the following formula:

for some \(\xi\) in \((a, b)\). This estimation indicates that the error decreases quadratically as the number of subintervals \(n\) increases, assuming \(f''(x)\) is bounded and continuous.

6. Applications of the Trapezoidal Rule

The trapezoidal rule is widely used in various fields, including physics, engineering, economics, and biology, for tasks such as:

• Calculating areas under curves when an antiderivative is difficult to find.
• Estimating distances traveled over time with varying velocities.
• Approximating solutions to differential equations.

Its simplicity and improved accuracy over basic Riemann sums make it a valuable tool for both theoretical studies and practical applications.

7. Example Problem

Problem: Use the trapezoidal rule with \(n = 4\) to approximate the integral:

Solution:

1. Determine \(\Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5\).
2. Identify the endpoints: \(x_0 = 0\), \(x_1 = 0.5\), \(x_2 = 1.0\), \(x_3 = 1.5\), \(x_4 = 2.0\).
3. Calculate the function values:
   • \(f(x_0) = f(0) = 0^2 + 1 = 1\)
   • \(f(x_1) = f(0.5) = (0.5)^2 + 1 = 1.25\)
   • \(f(x_2) = f(1.0) = 1^2 + 1 = 2\)
   • \(f(x_3) = f(1.5) = (1.5)^2 + 1 = 3.25\)
   • \(f(x_4) = f(2.0) = 2^2 + 1 = 5\)
4. Apply the trapezoidal rule: $$T_4 = \frac{0.5}{2} [1 + 2(1.25 + 2 + 3.25) + 5] = 0.25 [1 + 2(6.5) + 5] = 0.25 [1 + 13 + 5] = 0.25 \times 19 = 4.75$$
5. The exact value of the integral is: $$\int_{0}^{2} (x^2 + 1) \, dx = \left[\frac{x^3}{3} + x\right]_{0}^{2} = \left(\frac{8}{3} + 2\right) - 0 = \frac{14}{3} \approx 4.6667$$
6. The trapezoidal approximation \(T_4 = 4.75\) is close to the exact value \(4.6667\), demonstrating the method's accuracy.

8. Advantages of the Trapezoidal Rule

• Improved accuracy over basic Riemann sums due to the use of trapezoids.
• Simplicity in computation, making it accessible for manual calculations.
• Applicability to a wide range of functions, including those without elementary antiderivatives.

9. Limitations of the Trapezoidal Rule

• Accuracy decreases for functions with high curvature unless a large number of subintervals are used.
• May not be suitable for functions with discontinuities or singularities within the interval.
• Requires knowledge of the function's second derivative for error estimation.

10. Trapezoidal Rule vs. Other Numerical Integration Methods

While the trapezoidal rule offers a balance between simplicity and accuracy, other methods like Simpson's rule can provide higher accuracy with fewer subintervals by using quadratic approximations. However, Simpson's rule requires the number of subintervals to be even and may involve more complex computations.

Comparison Table

Summary and Key Takeaways