1. What is Simpson's Rule?

Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses quadratic polynomials to approximate the function 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:

Where:

• \( I \) — Integral approximation
• \( h \) — Step size = \( \frac{b-a}{n} \)
• \( y_i \) — Function values at points \( x_i = a + i \times h \)
• \( n \) — Number of intervals (must be even)

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

3. Importance of Numerical Integration

Details: Simpson's Rule is essential for calculating definite integrals when an analytical solution is difficult or impossible to obtain. It's widely used in engineering, physics, and computational mathematics.

4. Using the Calculator

Tips: Enter the lower and upper limits of integration, an even number of intervals, and the function f(x) 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 the number of intervals be even?
A: Simpson's Rule requires pairs of subintervals to fit parabolas, so n must be even to ensure proper partitioning.

Q2: How accurate is Simpson's Rule?
A: Simpson's Rule has an error term proportional to h⁴, making it more accurate than trapezoidal rule (error ∝ h²) for smooth functions.

Q3: What types of functions work best?
A: Works best with smooth, continuous functions. Accuracy decreases for functions with discontinuities or sharp changes.

Q4: Can I use trigonometric functions?
A: Yes, the calculator supports standard mathematical functions including sin(x), cos(x), tan(x), etc.

Q5: What if I get an error?
A: Check that your function is properly formatted and that the number of intervals is even and ≥2.