Simpson's Rule Formula Calculator Tool

Simpson's Rule Formula:

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n) \right] \]

where \( h = \frac{b - a}{n} \) and \( n \) is even

Approximate Integral:

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1. What is Simpson's Rule?

Simpson's Rule is a numerical method for approximating definite integrals using quadratic polynomials. It provides more accurate results than simpler methods like the trapezoidal rule by fitting parabolas to sections of the curve.

2. How Does the Calculator Work?

The calculator uses the Simpson's Rule formula:

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n) \right] \]

where \( h = \frac{b - a}{n} \) and \( n \) is even

Where:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Even number of subintervals
\( h \) — Width of each subinterval
\( f(x_i) \) — Function values at each point

Explanation: The method weights the function values with alternating coefficients of 4 and 2 (except first and last) to achieve higher accuracy through quadratic approximation.

3. Importance of Numerical Integration

Details: Simpson's Rule is essential when dealing with functions that are difficult or impossible to integrate analytically, or when working with discrete data points.

4. Using the Calculator

Tips: Enter the lower and upper limits, an even number of subintervals, and comma-separated function values. The number of function values must be exactly n+1.

5. Frequently Asked Questions (FAQ)

Q1: Why must n be even?
A: Simpson's Rule works by fitting parabolas to pairs of subintervals, which requires an even number of divisions.

Q2: How accurate is Simpson's Rule?
A: It's generally more accurate than the trapezoidal rule, with error proportional to the fourth power of the subinterval width.

Q3: When should I use Simpson's Rule?
A: Use it when you need more accurate results than the trapezoidal rule provides, especially for smooth functions.

Q4: What if I have an odd number of points?
A: You can either use the trapezoidal rule for the last interval or reduce n by 1 to make it even.

Q5: Can I use this for experimental data?
A: Yes, as long as you have equally spaced data points and an even number of intervals.