Simpson's Rule Integral Calculator

Simpson's Rule:

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

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 for smooth functions.

2. How Does the Calculator Work?

The calculator uses Simpson's Rule formula:

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

Where:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Even number of subintervals
\( h \) — Width of each subinterval
\( f(x) \) — Function to integrate

Explanation: The method approximates the area under the curve by fitting parabolas to successive triplets of points.

3. Importance of Numerical Integration

Details: Numerical integration is essential when functions cannot be integrated analytically or when dealing with empirical data. Simpson's Rule provides good accuracy for smooth functions with relatively few evaluations.

4. Using the Calculator

Tips: Enter valid limits (b > a), an even number of subintervals (n ≥ 2), and a properly formatted function (use 'x' as variable, ^ for powers, standard math operations).

5. Frequently Asked Questions (FAQ)

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

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

Q3: What functions work best?
A: Works best with smooth functions. Less accurate for functions with discontinuities or sharp changes.

Q4: Can I use trigonometric functions?
A: Yes, but you would need to modify the calculator to support them (current version supports basic operations).

Q5: What's the advantage over other methods?
A: For the same computational effort, Simpson's Rule typically gives more accurate results than simpler methods.