The Simpson's Rule Calculator is a handy tool used in mathematics and numerical analysis to approximate the integral of a given function over a specified interval. It employs a method known as Simpson's Rule, which provides a more accurate estimation compared to simple methods like the trapezoidal rule.
Formula of Simpson's Rule Calculator
The Simpson's Rule Calculator utilizes the following formula:
S = (h/3) * [f(x0) + 4∑f(xi) + 2∑f(xi+1) + f(xn)]
Where:
- S is the approximation of the integral.
- h is the width of each subinterval, given by (b - a) / n, where a and b are the lower and upper limits of integration, respectively, and n is the number of subintervals.
- ∑ denotes summation.
- xi represents the values of x within the interval [a, b] at which the function f(x) is evaluated.
Table of General Terms
| Term | Description |
|---|---|
| Integral | A fundamental concept in calculus representing the area under a curve. |
| Approximation | An estimation of a value that is close to the actual value. |
| Subinterval | A smaller interval within a larger interval. |
| Numerical Analysis | The study of algorithms that use numerical approximation for solving problems. |
Example of Simpson's Rule Calculator
Let's consider an example to illustrate how the Simpson's Rule Calculator works. Suppose we want to find the integral of the function f(x) = x^2 from x = 0 to x = 2.
Using Simpson's Rule with, say, 4 subintervals:
- Calculate h = (2 - 0) / 4 = 0.5.
- Evaluate f(x) at each endpoint and midpoint of the subintervals.
- Apply Simpson's Rule formula to approximate the integral.
Most Common FAQs
Simpson's Rule is a numerical method for approximating the integral of a function using quadratic polynomials.
Simpson's Rule tends to provide more accurate approximations compared to simpler methods like the trapezoidal rule, especially when the function is smooth.
Simpson's Rule is useful when you need a more accurate approximation of an integral, particularly for functions with higher-order derivatives.
