1. What is the Midpoint Rule?

Definition: The Midpoint Rule is a numerical method for approximating definite integrals by dividing the area under a curve into rectangles.

Purpose: It provides a simple way to estimate integrals when an exact solution is difficult or impossible to find analytically.

2. How Does the Midpoint Rule Work?

The calculator uses the formula:

Where:

• \( I \) — Integral approximation
• \( \Delta x \) — Width of each interval (\( \frac{b-a}{n} \))
• \( f(x_i) \) — Function value at midpoint of interval i
• \( n \) — Number of intervals

Explanation: The area under the curve is approximated by summing the areas of rectangles whose heights are determined by the function value at the midpoint of each interval.

3. Importance of Numerical Integration

Details: Numerical integration is essential for solving real-world problems where functions cannot be integrated symbolically or when dealing with empirical data.

4. Using the Calculator

Tips: Enter the function (e.g., "x^2", "sin(x)", "exp(x)"), integration limits, and number of intervals. More intervals generally give better accuracy but require more computation.

5. Frequently Asked Questions (FAQ)

Q1: How accurate is the Midpoint Rule?
A: The error decreases as the number of intervals increases. It's generally more accurate than the Left or Right Endpoint rules.

Q2: When should I use more intervals?
A: Use more intervals when the function changes rapidly or when higher precision is needed.

Q3: What functions can I enter?
A: Basic arithmetic operations and common functions like sin, cos, exp, log. Use PHP syntax (e.g., x^2 should be written as pow(x,2) for full accuracy).

Q4: How does this compare to other methods?
A: The Midpoint Rule is more accurate than endpoint rules but less accurate than Simpson's Rule for the same number of intervals.

Q5: Can I use this for improper integrals?
A: No, this calculator is designed for proper integrals with finite limits.