Simpsons Formula 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 Rule?

Simpson's Rule is a numerical method for approximating definite integrals. It works by approximating the area under a curve using parabolic arcs, 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 \( h = \frac{b - a}{n} \) 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 approximates the integral by fitting parabolas to sections of the curve and calculating the area under these parabolas.

3. Importance of Numerical Integration

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

4. Using the Calculator

Tips: Enter the lower and upper limits, an even number of subintervals (more intervals = better accuracy but slower computation), and the function to integrate (using PHP math syntax, e.g., "sin(x)", "x^2 + 3*x - 2").

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 intervals.

Q2: How accurate is Simpson's Rule?
A: It's generally more accurate than the trapezoidal rule, with error proportional to \( h^4 \). Doubling n reduces error by about a factor of 16.

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 functions can I enter?
A: Any valid PHP mathematical expression using x as the variable (e.g., "exp(x)", "sqrt(x)", "x^3 + 2*x").

Q5: What if I get an error?
A: Check that your function syntax is correct and that n is even. Also ensure b > a.