Simpsons 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 area under a curve using quadratic polynomials (parabolic arcs) rather than straight lines (as in the Trapezoidal Rule), resulting in more accurate approximations.

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

Where:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Number of subintervals (must be even)
\( h \) — Width of each subinterval
\( f(x) \) — Function to integrate

Explanation: The method weights the function values in a specific pattern (1, 4, 2, 4, ..., 4, 1) to achieve higher accuracy than simpler methods.

3. Importance of Numerical Integration

Details: Numerical integration is essential when an integral cannot be solved analytically or when working with empirical data. Simpson's Rule provides a good balance between accuracy and computational complexity.

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 standard mathematical notation (e.g., x^2 for x², sin(x) for sine function).

5. Frequently Asked Questions (FAQ)

Q1: Why must n be even?
A: Simpson's Rule works by fitting parabolas to groups of three points, so we need an even number of intervals (odd number of points).

Q2: How accurate is Simpson's Rule?
A: The error is proportional to (b-a)h⁴, making it much more accurate than the Trapezoidal Rule for smooth functions.

Q3: What functions work best with Simpson's Rule?
A: It works best for smooth functions. For functions with discontinuities or sharp corners, other methods may be better.

Q4: Can I use this for improper integrals?
A: Not directly. Improper integrals require special treatment or transformation before numerical integration.

Q5: What's the difference between 1/3 and 3/8 rules?
A: The 1/3 rule uses quadratic polynomials (2nd order), while the 3/8 rule uses cubic polynomials (3rd order) and requires n divisible by 3.