Midpoint Rule Formula:
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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.
The calculator uses the Midpoint Rule formula:
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.
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.
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.
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.