Simpson's Method Calculator

Simpson's Rule:

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

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

Approximate Integral:

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1. What is Simpson's Method?

Simpson's Rule is a numerical integration technique that approximates the definite integral of a function by using quadratic polynomials (parabolic arcs) to approximate sections of the curve. It generally provides more accurate results than simpler methods like the trapezoidal rule.

2. How Does the Calculator Work?

The calculator uses Simpson's Rule:

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

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 evaluations in an alternating 4-2-4 pattern, which provides exact integration for cubic polynomials.

3. Importance of Numerical Integration

Details: Simpson's Rule is essential when dealing with functions that don't have elementary antiderivatives or when only discrete data points are available. It's widely used in engineering, physics, and computational sciences.

4. Using the Calculator

Tips:

Enter the lower and upper limits of integration
Specify an even number of subintervals (more intervals = better accuracy)
Enter the function using standard mathematical notation (e.g., sin(x), x^2, exp(x))
For best results with oscillating functions, ensure n is large enough to capture the oscillations

5. Frequently Asked Questions (FAQ)

Q1: Why must n be even?
A: Simpson's Rule works by fitting parabolas to pairs of subintervals, so it requires an even number of intervals to work properly.

Q2: How accurate is Simpson's Rule?
A: The error is proportional to (b-a)^5/n^4, making it much more accurate than trapezoidal rule for smooth functions.

Q3: When shouldn't I use Simpson's Rule?
A: Avoid it for functions with discontinuities or sharp corners, or when you don't have an even number of intervals.

Q4: What functions can I enter?
A: Most standard mathematical functions: sin, cos, tan, exp, log, sqrt, etc. Use x as the variable.

Q5: How many subintervals should I use?
A: Start with 10-20 for simple functions. For more complex functions or higher accuracy needs, use more.