1. What is the Trapezoidal Rule?

The Trapezoidal Rule is a numerical integration method that approximates the definite integral of a function by dividing the area under the curve into trapezoids. It provides a simple way to estimate integrals when an analytical solution is difficult or impossible to obtain.

2. How Does the Calculator Work?

The calculator uses the Trapezoidal Rule formula:

Where:

• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration
• \( f(a) \) — Function value at lower limit
• \( f(b) \) — Function value at upper limit

Explanation: The formula approximates the area under the curve between points a and b by treating it as a trapezoid.

3. Importance of Numerical Integration

Details: Numerical integration is essential in mathematics, physics, engineering, and many scientific fields where functions cannot be integrated analytically or when dealing with empirical data.

4. Using the Calculator

Tips: Enter the lower and upper limits of integration, and the function f(x) using standard mathematical notation (e.g., x^2, sin(x), exp(x)). Ensure b > a for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What is the accuracy of the Trapezoidal Rule?
A: The basic Trapezoidal Rule provides a first-order approximation. Accuracy improves when using multiple trapezoids (composite rule).

Q2: When should I use the Trapezoidal Rule?
A: Use it for simple integration problems or when you need a quick approximation. For more complex functions, consider Simpson's Rule or other methods.

Q3: What functions can I input?
A: You can input basic mathematical functions using standard notation (+, -, *, /, ^, sin, cos, tan, exp, log, etc.).

Q4: What are the limitations of this method?
A: The basic Trapezoidal Rule may be inaccurate for highly oscillatory functions or functions with rapid changes. The error is proportional to (b-a)³.

Q5: Can I use this for improper integrals?
A: The basic Trapezoidal Rule is designed for definite integrals over finite intervals. For improper integrals, specialized methods are required.