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Midpoint Rule Calculator

Midpoint Rule Formula:

\[ \int_{a}^{b} f(x) \,dx \approx \Delta x \left[f\left(\frac{x_0 + x_1}{2}\right) + f\left(\frac{x_1 + x_2}{2}\right) + \cdots + f\left(\frac{x_{n-1} + x_n}{2}\right)\right] \]

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1. What is the Midpoint Rule?

The Midpoint Rule is a numerical integration technique that approximates the definite integral of a function by dividing the area under the curve into rectangles and using the midpoint of each subinterval to determine the rectangle's height.

2. How Does the Calculator Work?

The calculator uses the Midpoint Rule formula:

\[ \int_{a}^{b} f(x) \,dx \approx \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right) \]

Where:

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 subinterval.

3. Importance of Numerical Integration

Details: Numerical integration is essential when an integral cannot be solved analytically or when dealing with empirical data. The Midpoint Rule provides a simple but effective approximation method.

4. Using the Calculator

Tips: Enter a valid mathematical function (using 'x' as variable), 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 Midpoint Rule has an error bound proportional to \( \frac{(b-a)^3}{24n^2} \), making it more accurate than the Left or Right Endpoint Rules for many functions.

Q2: When should I use more intervals?
A: Use more intervals when the function is highly variable or when you need greater precision. Start with 10-20 intervals and increase as needed.

Q3: What functions can I enter?
A: The calculator supports basic arithmetic operations (+, -, *, /), exponents (use ^), and common mathematical functions if properly implemented.

Q4: How does this compare to Simpson's Rule?
A: Simpson's Rule is generally more accurate as it uses parabolic approximations instead of rectangles, but requires more computation.

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

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