Symbolab Simpsons Rule Calculator

Simpson's Rule Formula:

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

Approximate Integral:

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1. What is Simpson's Rule?

Simpson's Rule is a numerical method for approximating definite integrals. It works by approximating the area under a curve using parabolic arcs between points, which typically provides better accuracy than simpler methods like the trapezoidal rule.

2. How Does the Calculator Work?

The calculator uses Simpson's Rule formula:

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

Where:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Even number of subintervals
\( h \) — Width of each subinterval
\( f(x_i) \) — Function value at point \( x_i \)

Explanation: The method weights function values differently depending on their position - endpoints have weight 1, odd-indexed points have weight 4, and even-indexed points have weight 2.

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 but slower computation), and the function to integrate. Use standard mathematical notation (e.g., x^2 for x squared).

5. Frequently Asked Questions (FAQ)

Q1: Why must n be even?
A: Simpson's Rule works by fitting parabolas to each pair of subintervals, so it 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 (b-a)^5/n^4.

Q3: What functions can I enter?
A: This calculator supports basic operations (+, -, *, /, ^) and common functions (sin, cos, tan, exp, log). For complex functions, consider specialized software.

Q4: When shouldn't I use Simpson's Rule?
A: Avoid when the function is highly oscillatory or discontinuous, or when you need very high precision.

Q5: Can I use decimal values for limits?
A: Yes, the calculator supports decimal values for all numerical inputs.