Simpson's Rule Calculator

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. It works by approximating the integrand with quadratic polynomials (parabolas) over subintervals, providing more accurate results than simpler methods like the trapezoidal rule.

2. How Does the Calculator Work?

The calculator uses 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:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Even number of subintervals
\( h = \frac{b - a}{n} \) — Width of each subinterval
\( x_i = a + ih \) — Points where function is evaluated

Explanation: The method weights function values with coefficients 1, 4, 2, 4, ..., 4, 1 to achieve higher accuracy than simpler methods.

3. Importance of Numerical Integration

Details: Simpson's Rule is essential when integrals cannot be solved analytically or when working with empirical data. It's widely used in physics, engineering, and computational mathematics.

4. Using the Calculator

Tips: Enter the lower and upper limits, an even number of subintervals (more intervals = better accuracy), and the function to integrate (use "x" as variable).

5. Frequently Asked Questions (FAQ)

Q1: Why must n be even?
A: Simpson's Rule pairs intervals to fit parabolas, requiring an even number of subintervals.

Q2: How accurate is Simpson's Rule?
A: It's exact for cubic polynomials and generally more accurate than trapezoidal rule for the same n.

Q3: What functions can I enter?
A: Basic math operations (+, -, *, /), powers (^), and functions like sin, cos, exp, log, etc.

Q4: When should I use more subintervals?
A: For rapidly changing functions or when higher precision is needed. Double n to roughly quadruple accuracy.

Q5: Are there limitations?
A: Not suitable for discontinuous functions or improper integrals. Adaptive methods may be better for some cases.