Trapezoidal Rule Integration Method

Overview of Integral Calculus

Integral Calculus: A key branch of mathematics focused on integration, essential for determining areas under curves. It has applications across various fields, including mathematics, physics, electrical engineering, and economics.

Key Definitions

• Indefinite Integral: The antiderivative of a function, expressed as $\int f(x) \, dx = F(x) + C$.
• Definite Integral: Computes the exact area under a curve between the limits $a$ and $b$, represented as $\int_{a}^{b} f(x) \, dx$.

Introduction to the Trapezoidal Rule

Definition and Purpose

• Trapezoidal Rule: A numerical approach to approximate definite integrals utilising trapezoids.
• Essential in numerical integration when exact solutions are unavailable.
• Offers a practical method for estimating areas beneath curves.

Geometric Reasoning

• Why Trapezoids?
  • Trapezoids align more closely with curves than rectangles, significantly minimising approximation errors.

Definition of Key Components

• Integral Limits: The start and end points ($a$ and $b$).
• Endpoint Function Values: Evaluated at the boundaries of the interval, $f(a)$ and $f(b)$.
• Intermediate Function Values: Evaluated at subinterval points, contributing to enhanced accuracy.
• Subintervals: Uniform divisions that improve precision.

Formula of the Trapezoidal Rule

The Trapezoidal Rule is expressed by the formula:
$\int_a^b f(x) \, dx \approx \frac{b-a}{2n} \left[f(a) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(b) \right]$

Worked Example and Derivation

Derivation Process

• Decompose the Interval: Divide the curve into equidistant trapezoids.
• Calculate Areas: Determine each area's size using the height $(b-a)/n$ and a sum-based base value.
• Develop Formula through geometric analysis.

Example Calculation

Example: Approximate $\int_0^1 x^2 \, dx$ using $n = 2$.

• Step 1: Compute the width: $(b-a)/n = 0.5$.
• Step 2: Evaluate the function values: $f(0) = 0$, $f(0.5) = 0.25$, $f(1) = 1$.
• Step 3: Apply the formula:
  $\int_0^1 x^2 \, dx \approx \frac{0.5}{2} \left[0 + 2 \times 0.25 + 1 \right] = 0.375$

Practice Problem with Solution

Problem: Calculate $\int_0^\pi \sin x \, dx$ using $n = 4$.

Solution:

• Width: $(b-a)/n = \pi/4$
• Function values:
  • $f(0) = \sin(0) = 0$
  • $f(\pi/4) = \sin(\pi/4) = 1/\sqrt{2} \approx 0.7071$
  • $f(\pi/2) = \sin(\pi/2) = 1$
  • $f(3\pi/4) = \sin(3\pi/4) = 1/\sqrt{2} \approx 0.7071$
  • $f(\pi) = \sin(\pi) = 0$
• Apply the formula: $\int_0^\pi \sin x \, dx \approx \frac{\pi/4}{2}[0 + 2(0.7071) + 2(1) + 2(0.7071) + 0] = \frac{\pi}{8}[4.8284] \approx 1.8954$

The exact value is 2, so our approximation is quite close.

Practical Applications

Land Measurement

• Vital in Civil Engineering for estimating areas of irregular plots.
• Example: With bases $b_1 = 50$ m and $b_2 = 70$ m, height $h = 30$ m. Area = 1800 sq. m.

Volume Calculations

• Applied in Science and Engineering to estimate tank and container volumes.
• Example: Tank with bases $b_1 = 10$ m, $b_2 = 15$ m, height $h = 20$ m. Volume = 250 m³.

Addressing Common Misconceptions

Misconception 1: Exact Results

• The Trapezoidal Rule provides approximate, not exact, results.

Misconception 2: Weights of Endpoint and Interior Values

• Endpoint values have less significance than interior ones.

By understanding the simplicity and limitations of the Trapezoidal Rule, students can effectively approximate integrals and apply these techniques across various disciplines.