1. What is the Midpoint Rule?

Definition: The Midpoint Rule is a numerical method for approximating definite integrals by summing function values at interval midpoints.

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 formula used is:

Where:

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

Explanation: The area under the curve is approximated by rectangles with heights equal to the function value at each interval's midpoint.

3. Importance of the Midpoint Rule

Details: This method provides better accuracy than the left or right endpoint rules for the same number of intervals, especially for smooth functions.

4. Using the Calculator

Tips: Enter the function (e.g., "x^2", "sin(x)"), integration limits, and number of intervals. More intervals 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 square of the number of intervals, making it more accurate than endpoint rules for most functions.

Q2: What functions work best with this method?
A: The Midpoint Rule works well for smooth, continuous functions. It's less accurate for functions with rapid changes or discontinuities.

Q3: How do I choose the number of intervals?
A: Start with 4-10 intervals for simple functions. For better accuracy, increase the number (e.g., 100-1000).

Q4: Can I use this for improper integrals?
A: No, this calculator is designed for standard definite integrals with finite limits.

Q5: How does this compare to other numerical methods?
A: The Midpoint Rule is more accurate than the Trapezoidal Rule for the same number of intervals but less accurate than Simpson's Rule.